mysql_get_server_info(): supplied argument is not a valid MySQL-Link resource

Hi,
I launched a new Wordpress multi-site. At times when the server gets up to max capacity I get very ugly error messages.

Like mysql_get_server_info(): supplied argument is not a valid MySQL-Link resource

And Error connecting to database....

It's all going on in db.php

How to trap this earlier and produce a NICE message, instead of this horrible crap....