Right. This sounds very much like a hardware problem, in fact I'm virtually certain
of it and most likely it is the CPU overheating. First thing to check is that all
the cooling fans are turning and that the machine is generally free of dust/fluff
etc. Desktops are rather easier to get apart for cleaning than laptops. A paint brush
and vacuum cleaner are the tools needed for the job. Also worth doing at the same
time is to re-seat the memory modules by just unclipping them and pressing them straight
back in again, to ensure that all the edge-contacts are making a good connection.
Hope this helps. Cheers. Spexx.