Abit BP6 system will not POST

[owen at nerdnetworks.org (Owen B. Mehegan)]

A few days ago, I woke up and found my computer frozen. I reset it, and the
system would not come back up. The drives would spin, but the system did not
beep and wouldn't POST. Eventually, I took out one CPU, and the box seemed
to boot fine. Awhile later, however, it crashed again, and would not reboot.
After a few more tries, I got it to load and run the memory test, and then
it failed again. Sometimes the system does not even POST. Other times it
stops running/freezes after the memory test, during the OS loading process,
or in the BIOS settings menus - NEVER seen that happen before.

So far, I have pulled one CPU, re-seated the video card, cleared the CMOS,
and swapped/removed RAM from the system, all without success. Up until now,
the system has run fine, and I've made no recent changes to the hardware.
The problem sounds like it could be heat, but even this morning, after the
system had been off overnight, it would not even POST. Pertinent specs:

Abit BP6 motherboard
Dual Celeron 533 processors (not overclocked)
2x 256M Crucial PC 133 memory

Any suggestions as to what could be causing this problem would be much
appreciated. Thanks!

/owen