MySQL - Connection blocked because of many connection errors - How to find root cause?
The advice you've seen about max_connections is a distraction from the issue you're seeing, because it has nothing to do with connection errors.

You can log the failed connections to the mysql error log by issuing this:

SET GLOBAL log_warnings = 2;


You can also make this a permanent setting in my.cnf so that it persists after a restart.

http://dev.mysql.com/doc/refman/5.5/en/server-options.html#option_mysqld_log-warnings