Is the CPU problem?

I'd first try another known good ATX power supply slaved in...power supplies do occasionally die, and perhaps the voltages on your existing one are maginal to start off with, resulting in intermittent operation...

Intermitant start is usually a power issue.
Things to check include the power supply, but also include the power cord (make sure bothe ends are plugged in and carry voltage) the power supply to mobo connector (look for a loose connection, or black marks on the pins), or a broken case switch. (take the wire off the mobo, and use a continuity or ohm tester to make sure the case switch works well. If you dont have access to a tester, use a jumper across the pins, to see if the system starts without problem, a bunch of times).
If those things give no joy, it could be something else loose, like gfx card, nic card, or ram.

Well if it's still covered under warranty, then that's not a bad suggestion. They'll repair it with parts equally crappy to the ones they built it with that caused the problem in the first place, but it might buy you six months to a year of trouble-free computing. I only reccomend doing things yourself once the warranty period is over. But it's your computer and your choice. I'm only making suggestions. Chances are it's really just the power supply. In fact, it might even be as simple as dust.