Then after that, I would start looking at heat. Is the computer free of heat trapping dust? Do all fans turn freely?
While in there, inspect the motherboard for bulging or leaking electrolytic capacitors. These failed or failing capacitors are a common cause of sudden, but seemingly random system lock ups and reboots.
The capacitors look like tall soda cans, many of which surround the CPU socket. All older and most newer motherboards use electrolytic capacitors, which contain a
liquid electrolyte. Failing (including flawed and/or abused/over-heated) capacitors literally bulge at the seams due to excessive internal pressures. Extreme (and very rare) cases result in a firecracker type explosion that can really stink up a room.
Typically, electrolyte just
oozes from the pressure relief points stamped in the tops of the capacitor casings (seen as a symbol or letter). The electrolyte can be caustic to motherboards and flesh.
Look for white to dark-brown, dried liquid or foam on the tops or bottoms of the capacitors. Bulging capacitors are a sign leakage is about to occur.
A stressed, underpowered or failing power supply can cause some very odd problems, including reboots.
You can test your RAM using one of the following programs. Both require you to create and boot to a bootable floppy disk or CD to run the diagnostics. Using the floppy method is generally easier and yet another reason I still include floppy drives in my new PC builds. However, the CD method is just as effective at detecting RAM problems. Allow the diagnostics to run for several passes or even overnight. You should have
no reported errors.
Windows Memory Diagnostic - see the easy to follow instructions under Quick Start Information.
or
MemTest86+ (for more advanced users) - an excellent how-to guide is available here.