Hmm, I just remembered something. Lots of machines between 2001 and 2003 had leaking cap problems... Typical symptoms would be random shutdowns for a period and then the system would stop booting at all. A quick check you can do is to look inside your system at the motherboard and look for cylinder shape objects that stick up. There should be tons of them. Those are capacitors. The top metal portion of the caps should be a flat surface, but occasionally they are manufactured incorrectly and can turn bad.
You can bump into either bulging or leaking caps. Here is an image to help you reference:
In this small area you can see a bunch of caps. The one near the ATX power connector that has crust and stuff all on top... that's one that has blown. They leak a battery acid-like substance on the motherboard or fail to hold charges like they are supposed to and usually it's your CPU voltage that suffers. If any of the cylinders in your PC have brown crusty stuff on top, then you've got blown caps, and that means the motherboard is bad. Also, some might just be bulged, meaning they aren't flat on top.
More info and pics:
http://badcaps.net/ident/
This is a VERY common problem with PC's dating back a few years and several companies have settled class action lawsuits because of this problem.
If you have bad capacitors, you might as well just pronounce the machine dead. Fixing the caps is by no means an easy task though it can be done and parts are obtainable at Radio Shack. However when you consider the labor, part costs, and risks of failure... it's worth it to buy a new motherboard.
There's nowhere I can't reach.