[GENERAL] sorry, too many clients already error

I have set max_connection = 40.

The usage is somewhat not typical. It is basically
experiment runs that connect to the database
and dump results there.

The experiments connect through JDBC and
they close the connection when they are done.

I can verify that no more than 20 clients/experiments
are active at any given time.

What could be the cause of the database issuing
too many clients error. Could I have set resources too high?

Thanks.
-sandeep