I had that EXACT same problem on my 1.2 GHz thunderbird system for about half a year. I still don't know for sure what caused it. But the only thing I ever replaced was the PSU, and it stopped happening, but I don't remember if that fixed it immediately or if the problem was actually heat, because I was frequently tinkering with my one-fan case, trying to get it cooler. Here's a suggestion: Open both sides of your case and put a box fan (the big stand up type) a few inches to the left of your system. See if that allows it to run.

Take note that this problem occured during the hotter parts of the year, which is why I attributed it to heat/humidity.