Have you tried running your computer with the case side removed? This way you can check that your CPU fan is working properly. It maybe that your CPU is overheating.
Personally, in this situation, I would purchase a decent heatsink and fan - making sure there is sufficient gel between the CPU and heatsink. However, this may not be the problem.
I have recently had a computer with a similar problem - but mine did not reboot and the fault lay with the PSU. Never mind, this will bump you to the top again. :-)