It sounds to me like you are on a shared host. Is it possible that someone else on the same server is using up the available connections to the mySQL server?

Generally, even if you do not close all of your connections properly the PHP processor will close them as soon as the page has been processed and sent to the server for delivery. Pages typically take less than half a second to be processed and even moderately busy servers are typically handling only 10 pages at any given moment. 10 connections should not exceed your SQL limit unless the hosting company has set some ridiculously low limits or someone else on your server is doing something they shouldn't and connections are not being throttled on a per-account basis.

__________________
The best way to learn anything, is to question everything.