Just don't turn it off!
I just got a pc from a friend of mine that does something strange...if you turn it off, it won't come back on for a half hr or so. If you restart, it's fine but if it gets shutdown (or the power goes out like happened last week...), it usually won't come back on for a while. The power light on the mb goes off and everything. That's how I tell when it's ready to come back on...lol...I look through the case window and when the mb light is back on, I know I'm good to go. Then it works perfectly fine.
Only thing I can think of is that the psu is on the outs. Agreed? Seems like quite a strange symptom for a failing psu.
Oh...and, if it matters, it's an intel core 2 quad, asus (p5n-d, I want to say) mb, 4 gb ram...not sure what kind of psu.