My power supply/motherboard nightmare
Solution 1:
What you described to me looks like some sort of safety system in the power supply.
I recommend that you disassemble whole computer and check for any burn marks on the motherboard and pretty much everything else. There could be something which causes a short and makes power supply turn off. Pay close attention to computer case and check if it's moving or vibrating.
Next step would be to check cables in your power supply. If you have multi-rail power supply, you could be hitting rail limit for power. Make sure that load is distributed among cables and devices. If a cable has several connectors on it, try to use up all other cables first before using extra connectors. Make sure that you balance load on cables with several connectors. For example on a cable going into a hard drive make sure you only connect low power consumers and so on. This in general shouldn't be a problem, but some power supplies can be picky and may have bad load distribution.
Next check how different devices are mounted. I once had a similar problem. I bought a new computer and every once in a while it would shut down unexpectedly. In the end it turned out that the incompetent or malicious manufacturer used too long screw on the floppy drive. They penetrated inside and caused shorts from time to time. Make sure that nothing similar is happening in your case.
Solution 2:
The more I think about this the more I think it is the BIOS turning off the system due to CPU overheat, I had exactly the same problem once.
Look in the BIOS and see if it has a setting that sets the temperature at which it shuts off.
You can force the issue with cpuburn (google it) there are different versions for different CPUs, choose the one for your cpu, It will cause the CPU to heat up rapidly and if it triggers the shutdown then there is a good chance that is the problem.
Check the BIOS to see what temp it is set to shutdown at, and check the CPU specs, it may be set too low, if not you will need a better CPU cooler, or a bigger fan blowing on the CPU cooler.