There are a couple of things you can do to work around the connections limitation. First, you can find this error if your scripts open persistent connections, wich aren't closed even if the script terminates. Use mysql_connect() instead of mysql_pconnect() unless you have a good reason. In particular, check this setting in third-party scripts (such as osCommerce).
Next, the file /etc/my.cnf typically holds defaults for mysql. Note: You will need to login in the shell as root to make the changes. A line of
will increase the number of connections allowed to 500. This should be more than enough. Please note that the instruction should be placed in the [mysqld] section. Otherwise MySQL will ignore it.
After resetting this, restart mysql (you would run "service mysqld restart" as root or issue the MySQL restart via WHM).
Now you should never run out of mysql connections.