Hi,
this could be a problem of mysql_pconnect, since using mysql_pconnect keeps mysql connections open. To find the reason why they aren't closed is a little bit hard (heavy load, sites that use frames hosted on a server farm with a load balancer ....) but you might try to use mysql_connect instead since these connections normally get closed automatically after the php code has been parsed.
I had such a problem with a frame based site that was hosted on a server farm using a load balancer. The load balancer sent the requests for each frame to different servers thus opening a bunch of persistent connections at a time. Switching to mysql_connect solved the problem immediately.
This will not solve the problem if you have e.g. infinite loops in your code that contain mysql queries. That would force a connection to stay open forever (and bring the server down very fast, so this shouldn't be a problem here).
Thomas