I had that exact problem - running a 900MHz Duron. I didn't observe static handling precautions and damaged it while installing it. It worked fine until a minor storm. I replaced my RAM (which was also bad) but the CPU still caused random reboots/STOP errors. (CPU issues will also cause memory to show as bad in Memtest86).
Replaced with a 1.2ghz CPU - all was well. Except, I had to recap the motherboard. Then I threw it away
But no - if you run out of physical memory, things will just slow down worse and worse, never crash.