The error you’re receiving is a client-side error because the MySQL server cannot be reached.
The most likely candidate, in my opinion, is that your web servers cannot communicate with the domain on which the MySQL server resides. It could be DNS related too, but I really don’t know.
Can you ping the MySQL server from any of your web servers? That’s a good place to start.
This page might also provide you with some help. http://dev.mysql.com/doc/refman/5.0/en/can-not-connect-to-server.html