Try checking the case for dust and dirt. If the CPU heatsink is full of dust, the BIOS may be shutting down on overheat.
Clean the MB, cards, and CPU cooler with a 25mm long-bristle paintbrush and a small vacuum cleaner. Disassemble the (unplugged!) power supply and clean it out as well, and clean all the fans well. The dust I have seen in some former problem systems is atrocious.
While you are cleaning the fans, insure that they spin freely. Replace any that seem stiff. (You can often free them up by removing the sticker and bearing seal cap, and adding some 3 in 1 or Marvel Mystery oil, but sometimes a stiff fan has already wiped its bearings and will get noisy and stiff again quickly.)
if cleaning removes a lot of dust, this may reduce your lockup/restart problems.