MySQL failed connection attempts hitting max_connections
I don't think the failed connections are causing the max connections issue, but can you set log_warnings=2 and after a short time, view the error log to see if there is more information there? The error log would record something like "access denied for user 'user'@'host'..."

The command to set log_warnings=2 is:

set global log_warnings=2;


You can find the error log like this:

show global variables like 'log_error';


What version of mysql are you running? 

Can you also show the output of:

show global variables like 'skip_name_resolve';


and:

show global status like 'max_used_connections';