I would also try another power supply, and while inside, 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 motherboards, and many of today's less expensive motherboards use electrolytic capacitors containing 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 point, which appears as a symbol or letter stamped in the top of the capacitor casing. 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 motherboard with bulging or leaky capacitors can be repaired, but often it is more cost effective in the long run to replace the motherboard.
Be sure to first power down, unplug the computer, and keep yourself discharged by touching the bare metal of the case before reaching in.