I experimented with it a bit. Doing things with CPU or video did not seem to be affecting anything. I ran this batch script alongside playing an HD movie:
This caused CPU (both cores) to go to 80% utilization. Nothing. But the moment I copied the data, the power would go off.
So today I went to Fry's and bought a new motherboard, a thermal compound, and a Cooler Master Real Power Pro 750W power supply.
The power supply I bought mostly because it was unbelievably cheap - $125 - $50 rebate = $75. Deals like these do not happen very often, even at Fry's. [The deal is not on the web site by the way - boo! http://www.frys-electronics-ads.com/]
I did not actually expect that it was a power supply that was going bad because my CPU/video test was working.
However, because changing the power supply is much-much-much easier than changing the motherboard (I was really not looking forward to poring through my records trying to match Windows key to the hardware which was using it), I decided to try that first.
So I did that, and no more shutdowns. The old one was 450W, but it is not a very power-hungry system - 65W CPU, only one hard drive, and the video is a mid-range 7600GS Nvidia card. Be it as it may, the problem seems to be solved.
But why maxing out video card and the CPU was working, whereas disk/network was not - beats me. I'd love to know the answer though.