diff --git a/mypy/build.py b/mypy/build.py
--- a/mypy/build.py
+++ b/mypy/build.py
@@ -224,7 +224,8 @@ def _build(sources: List[BuildSource],
                     lambda path: read_py_file(path, cached_read, options.python_version),
                     options.show_absolute_path,
                     options.enabled_error_codes,
-                    options.disabled_error_codes)
+                    options.disabled_error_codes,
+                    options.many_errors_threshold)
     plugin, snapshot = load_plugins(options, errors, stdout, extra_plugins)
 
     # Add catch-all .gitignore to cache dir if we created it
diff --git a/mypy/defaults.py b/mypy/defaults.py
--- a/mypy/defaults.py
+++ b/mypy/defaults.py
@@ -30,3 +30,7 @@
                   'html',
                   'txt',
                   'lineprecision']  # type: Final
+
+# Threshold after which we sometimes filter out most errors to avoid very
+# verbose output
+MANY_ERRORS_THRESHOLD = 200  # type: Final
diff --git a/mypy/errors.py b/mypy/errors.py
--- a/mypy/errors.py
+++ b/mypy/errors.py
@@ -10,7 +10,7 @@
 from mypy.scope import Scope
 from mypy.options import Options
 from mypy.version import __version__ as mypy_version
-from mypy.errorcodes import ErrorCode
+from mypy.errorcodes import ErrorCode, IMPORT
 from mypy import errorcodes as codes
 from mypy.util import DEFAULT_SOURCE_OFFSET, is_typeshed_file
 
@@ -65,6 +65,10 @@ class ErrorInfo:
     # Fine-grained incremental target where this was reported
     target = None  # type: Optional[str]
 
+    # If True, don't show this message in output, but still record the error (needed
+    # by mypy daemon)
+    hidden = False
+
     def __init__(self,
                  import_ctx: List[Tuple[str, int]],
                  file: str,
@@ -158,6 +162,10 @@ class Errors:
     target_module = None  # type: Optional[str]
     scope = None  # type: Optional[Scope]
 
+    # Have we seen an import-related error so far? If yes, we filter out other messages
+    # in some cases to avoid reporting huge numbers of errors.
+    seen_import_error = False
+
     def __init__(self,
                  show_error_context: bool = False,
                  show_column_numbers: bool = False,
@@ -166,7 +174,8 @@ def __init__(self,
                  read_source: Optional[Callable[[str], Optional[List[str]]]] = None,
                  show_absolute_path: bool = False,
                  enabled_error_codes: Optional[Set[ErrorCode]] = None,
-                 disabled_error_codes: Optional[Set[ErrorCode]] = None) -> None:
+                 disabled_error_codes: Optional[Set[ErrorCode]] = None,
+                 many_errors_threshold: int = -1) -> None:
         self.show_error_context = show_error_context
         self.show_column_numbers = show_column_numbers
         self.show_error_codes = show_error_codes
@@ -176,6 +185,7 @@ def __init__(self,
         self.read_source = read_source
         self.enabled_error_codes = enabled_error_codes or set()
         self.disabled_error_codes = disabled_error_codes or set()
+        self.many_errors_threshold = many_errors_threshold
         self.initialize()
 
     def initialize(self) -> None:
@@ -189,6 +199,7 @@ def initialize(self) -> None:
         self.only_once_messages = set()
         self.scope = None
         self.target_module = None
+        self.seen_import_error = False
 
     def reset(self) -> None:
         self.initialize()
@@ -201,12 +212,14 @@ def copy(self) -> 'Errors':
                      self.read_source,
                      self.show_absolute_path,
                      self.enabled_error_codes,
-                     self.disabled_error_codes)
+                     self.disabled_error_codes,
+                     self.many_errors_threshold)
         new.file = self.file
         new.import_ctx = self.import_ctx[:]
         new.function_or_member = self.function_or_member[:]
         new.target_module = self.target_module
         new.scope = self.scope
+        new.seen_import_error = self.seen_import_error
         return new
 
     def total_errors(self) -> int:
@@ -330,6 +343,8 @@ def _add_error_info(self, file: str, info: ErrorInfo) -> None:
         if file not in self.error_info_map:
             self.error_info_map[file] = []
         self.error_info_map[file].append(info)
+        if info.code is IMPORT:
+            self.seen_import_error = True
 
     def add_error_info(self, info: ErrorInfo) -> None:
         file, line, end_line = info.origin
@@ -354,8 +369,52 @@ def add_error_info(self, info: ErrorInfo) -> None:
             if info.message in self.only_once_messages:
                 return
             self.only_once_messages.add(info.message)
+        if self.seen_import_error and info.code is not IMPORT and self.has_many_errors():
+            # Missing stubs can easily cause thousands of errors about
+            # Any types, especially when upgrading to mypy 0.900,
+            # which no longer bundles third-party library stubs. Avoid
+            # showing too many errors to make it easier to see
+            # import-related errors.
+            info.hidden = True
+            self.report_hidden_errors(info)
         self._add_error_info(file, info)
 
+    def has_many_errors(self) -> bool:
+        if self.many_errors_threshold < 0:
+            return False
+        if len(self.error_info_map) >= self.many_errors_threshold:
+            return True
+        if sum(len(errors)
+               for errors in self.error_info_map.values()) >= self.many_errors_threshold:
+            return True
+        return False
+
+    def report_hidden_errors(self, info: ErrorInfo) -> None:
+        message = (
+            '(Skipping most remaining errors due to unresolved imports or missing stubs; ' +
+            'fix these first)'
+        )
+        if message in self.only_once_messages:
+            return
+        self.only_once_messages.add(message)
+        new_info = ErrorInfo(
+            import_ctx=info.import_ctx,
+            file=info.file,
+            module=info.module,
+            typ=None,
+            function_or_member=None,
+            line=info.line,
+            column=info.line,
+            severity='note',
+            message=message,
+            code=None,
+            blocker=False,
+            only_once=True,
+            origin=info.origin,
+            target=info.target,
+        )
+        self._add_error_info(info.origin[0], new_info)
+
     def is_ignored_error(self, line: int, info: ErrorInfo, ignores: Dict[int, List[str]]) -> bool:
         if info.blocker:
             # Blocking errors can never be ignored
@@ -453,6 +512,7 @@ def format_messages(self, error_info: List[ErrorInfo],
         severity 'error').
         """
         a = []  # type: List[str]
+        error_info = [info for info in error_info if not info.hidden]
         errors = self.render_messages(self.sort_messages(error_info))
         errors = self.remove_duplicates(errors)
         for file, line, column, severity, message, code in errors:
diff --git a/mypy/main.py b/mypy/main.py
--- a/mypy/main.py
+++ b/mypy/main.py
@@ -663,6 +663,8 @@ def add_invertible_flag(flag: str,
     add_invertible_flag('--show-absolute-path', default=False,
                         help="Show absolute paths to files",
                         group=error_group)
+    error_group.add_argument('--soft-error-limit', default=defaults.MANY_ERRORS_THRESHOLD,
+                             type=int, dest="many_errors_threshold", help=argparse.SUPPRESS)
 
     incremental_group = parser.add_argument_group(
         title='Incremental mode',
diff --git a/mypy/options.py b/mypy/options.py
--- a/mypy/options.py
+++ b/mypy/options.py
@@ -295,6 +295,10 @@ def __init__(self) -> None:
         self.show_absolute_path = False  # type: bool
         # Install missing stub packages if True
         self.install_types = False
+        # When we encounter errors that may cause many additional errors,
+        # skip most errors after this many messages have been reported.
+        # -1 means unlimited.
+        self.many_errors_threshold = defaults.MANY_ERRORS_THRESHOLD
 
     # To avoid breaking plugin compatibility, keep providing new_semantic_analyzer
     @property

@JukkaL possibly stupid question, but is this already happening? I tried to create a small reproducible test where I imported some of these packages and did some basic type checking against their functions, but all I'm seeing is the stub errors.

I'm sure I'm either misunderstanding or doing something wrong, but can you at least point me in the right direction with a small reproducible example (i.e., one import stub error + one other mypy error off of that import) so I can build on it?

Here's a test I tried adding to `check-modules.test` to start with to get the errors and figure out where/how to limit them:

```
[case testLimitLegacyStubErrorVolume]
# flags: --config-file tmp/pyproject.toml
from typing import Dict

import certifi
import chardet

where = certifi.where()  # type: str
certifi.contents()  # type: str
chardet.detect(b'chardet')  # type: Dict[str, str]

[file pyproject.toml]
\[tool.mypy]
allow_redefinition = false
allow_untyped_globals = false
check_untyped_defs = true
disallow_any_decorated = true
disallow_any_explicit = true
disallow_any_expr = true
disallow_any_generics = true
disallow_any_unimported = true
disallow_incomplete_defs = true
disallow_subclassing_any = true
disallow_untyped_calls = true
disallow_untyped_decorators = true
disallow_untyped_defs = true
disallow_subclassing_any = true
follow_imports = 'normal'
ignore_missing_imports = false
local_partial_types = true
no_implicit_optional = true
no_implicit_reexport = true
no_strict_optional = false
strict = true
strict_equality = true
strict_optional = true
warn_no_return = true
warn_redundant_casts = true
warn_return_any = true
warn_unreachable = true
warn_unused_configs = true

[builtins fixtures/tuple.pyi]
[builtins fixtures/dict.pyi]
```

But all I see is the stub errors (even though I was expecting to see errors on the function calls where I would be assigning an implicit `Any` to the typed vars).
diff --git a/test-data/unit/check-modules.test b/test-data/unit/check-modules.test
--- a/test-data/unit/check-modules.test
+++ b/test-data/unit/check-modules.test
@@ -2982,3 +2982,85 @@ T = TypeVar("T")
 class F(M):
     x: C
     class C: ...
+
+[case testLimitLegacyStubErrorVolume]
+# flags: --disallow-any-expr --soft-error-limit=5
+import certifi # E: Cannot find implementation or library stub for module named "certifi" \
+               # N: See https://mypy.readthedocs.io/en/stable/running_mypy.html#missing-imports
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # N: (Skipping most remaining errors due to unresolved imports or missing stubs; fix these first)
+certifi.x
+certifi.x
+certifi.x
+certifi.x
+
+[case testDoNotLimitErrorVolumeIfNotImportErrors]
+# flags: --disallow-any-expr --soft-error-limit=5
+def f(): pass
+certifi = f()  # E: Expression has type "Any"
+1()  # E: "int" not callable
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+1()  # E: "int" not callable
+
+
+[case testDoNotLimitImportErrorVolume]
+# flags: --disallow-any-expr --soft-error-limit=3
+import xyz1  # E: Cannot find implementation or library stub for module named "xyz1" \
+             # N: See https://mypy.readthedocs.io/en/stable/running_mypy.html#missing-imports
+import xyz2  # E: Cannot find implementation or library stub for module named "xyz2"
+import xyz3  # E: Cannot find implementation or library stub for module named "xyz3"
+import xyz4  # E: Cannot find implementation or library stub for module named "xyz4"
+
+[case testUnlimitedStubErrorVolume]
+# flags: --disallow-any-expr --soft-error-limit=-1
+import certifi # E: Cannot find implementation or library stub for module named "certifi" \
+               # N: See https://mypy.readthedocs.io/en/stable/running_mypy.html#missing-imports
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"
+certifi.x  # E: Expression has type "Any"

Limit the number of generated errors if stubs are missing
If stubs for third-party libraries have not been installed, mypy may generate thousands of errors, making it hard to figure out what's going on. One reason for this is that missing stubs cause many `Any` types to be inferred, and these may trigger additional errors, depending on the strictness options being used.

One way to address this is to use some heuristic to silence errors that are *not* about missing stubs after a threshold. For example, if we've reported N errors about missing stubs and at least M errors overall, hide any remaining errors that are not about stubs.

This could be a significant issue for some users after the next mypy release (0.900), which unbundles stubs for third-party libraries. Thus users are likely to see errors about missing stubs as soon they update to mypy 0.900.

mypy/test/testcheck.py::TypeCheckSuite::testDoNotLimitImportErrorVolume mypy/test/testcheck.py::TypeCheckSuite::testLimitLegacyStubErrorVolume mypy/test/testcheck.py::TypeCheckSuite::testUnlimitedStubErrorVolume mypy/test/testcheck.py::TypeCheckSuite::testDoNotLimitErrorVolumeIfNotImportErrors