Can you give us some specifics about the machine. What is the maximum number of concurrent connections have you Apache set to accept?
If you resell services, a client may have a rogue CGI or PHP script that is -- probably inadvertantly -- sucking up resources. Not very likely, but certainly possible. (Or, maybe it's one of yours that has gone awry by accident.)
Actually, it turns out my server was hacked. Someone hacked in and installed "RH-Sharpe's rootkit", according to the EV1 support people.
It's not the end of the world as I already moved my big client off of the server, but it sucks for the small number of people who were still on there - EV1 isn't allows me access without doing a system restore on a separate drive and mounting my old drive. SIGH.