Sounds like overheating to me of either a motherboard component, your graphics card, memory, or your CPU, or one of those units breaking down.
I have also witnessed this behaviour on computers where the power supply unit is breaking down.
Please note with the steps below, if you are unsure at all about how to do them, please do not, and remember, as always, static is a computer's enemy. If you haven't got a static strap, at least touch the metal of the case before touching any component.
Steps I would follow (after each step see if problem still occurs):
1. Open computer, and ensure all cooling fans are running and that dust has not collected en masse on any of the internal components. Also smell PSU and other components for the "burnt component" smell.
2. Inspect all components closely to ensure capacitors on the components have not "popped" (On bad Caps, you'll see the top of the capacitor swelled or broken, with yellow or orange ooze coming out). With other components, look for scorching.
3. Swap out graphics card
4. Swap out the memory (or try modules one by one)
5. Swap out PSU
6. Swap out CPU
7. Swap out motherboard
All of these are much easier to do if you have a second computer that is similair, or have a mate with a similair rig.
Please supply type of CPU, type of graphics card running, type of PSU and type of motherboard to more clearly isolate the error.
If you are running an external graphics card such as an NVidia or ATI and have an onboard graphics card as well, remove the graphics card and run on onboard graphics to see if this fixes the problem.
Finally, buy yourself a good on line surge unit such as an APC unit or similair (with battery backup, those non-battery units do little). These do greatly improve a computer's life, and provide less problems in the mean time.