I don't want to count my eggs before they hatch, but now it seems to be working, for the moment.
Would dulled or tarnished slot contacts be a factor? I went to a computer repair shop yesterday to get some things:
an S-video out adapter, a DFP [precursor to DVI] cable, the power supply I was going to swap out, and some DeOxit spray.
I sprayed the AGP slot and the PCI slot underneath that, and it seems more stable. I don't seem to get any more problems.
I have had contact problems before, when I had my previous motherboard. A spray of another brand of contact cleaner did the job, until I had to take the video card out. When either the card edge or slot was exposed for some time, it seemed to oxidize enough to where I needed to spray it again.
EDIT: I was wrong.
I did have a Vanta card I bought three weeks ago (It works with all slots filled), and I also have an NV18 (geForce 4, but does not work with all slots filled) and they both seem to confirm the theory that it's the power supply that can't provide enough current.
I did have problems and instability using the NV18 before, but I previously thought the problem laid in the capacitors that I replaced. It wasn't. It works just fine by itself.
Also, while testing, when I reset once before, it got the POST MEMORY COUNT wrong! (My memory is at 640MB. With all slots filled in normal circumstances, that particular fluke error counted around 570 MB or so.)