My system is as follows (it's almost 4 years old now, btw. Don't laugh)
MSI K7T Turbo LE
AMD T-BirdC 1333
2 40 gig HDDs (Maxtor and a WD I think)
3 DIMMs of 256 mb PC133 RAM (3 slots on the mobo)
LG DVD writer
Some soundcard, some NIC, and a floppy drive
whatever 300W PSU I got with the case
When I do get it back on and displaying data, the temperature displayed in the BIOS is ~43C which makes me think it's not an overheat issue, but I can only run the system for a few minutes before it powers off again, whereas if I leave it be overnight I can get an hour or 2 out of it the next morning, which makes me think it is heat.
I've got an old Voodoo3 kicking around, too. Should I change out video card and PSU and maybe removing a DIMM of RAM before going through the hassle of cleaning off and re-applying thermal paste?