Your instincts are good... a PSU on the way out could well give this sort of problem. As windows boots, it will try to activate everything hooked up to the motherboard in turn, each drawing more power from the PSU.
You may want to go into the computer's BIOS to check temperatures... The behaviour may be consistent with overheating too.
If you have a spare PSU hanging around, a swap & test would be a quick way to diagnose the problem more reliably.
If it isn't the PSU, then its time to try booting safe mode with only the CPU (& fan), 1 stick of RAM and the hard disk (and graphics card if you dont have integrated graphics - use the integrated ones if you have both) and see if it boots, adding things one by one until you find your suspect