The “Too many connections” problem is a common issue with applications using excessive permissions (and those that grant said global permissions). MySQL will always grant a user with SUPER privileges access to a DB to investigate the problem with a SHOW PROCESSLIST and where you can check the limits. I however found the following.

mysql> show global variables like 'max_connections';
+-----------------+-------+
| Variable_name   | Value |
+-----------------+-------+
| max_connections | 2000  |
+-----------------+-------+
1 row in set (0.00 sec)

mysql> show global status like 'max%';
+----------------------+-------+
| Variable_name        | Value |
+----------------------+-------+
| Max_used_connections | 6637  |
+----------------------+-------+
1 row in set (0.00 sec)

How can the max_used_connection exceed max_connections? This is possible because you can dynamically change max_connections in a normal MySQL environment. However ,this is AWS RDS where you cannot change variables dynamically via mysql client. You can via other command line options but this has not happened. Furthermore, this server is using the defauly.mysql.5.5 parameter group to further validate the claim that it has not been changed.

I do not have an answer for the client in this case.

I would also add this as another ding on the usability of RDS in production environments. I was locked out of the DB for a long time, and with no visibility of what was going on. The only options were wait, or restart the server. RDS does not provide this level of visibility of the processlist using a privileged user that could see what was going on. Perhaps an interface they should consider in future.