it could be several things. Some system boards will cut things off if the system gets too hot (though usually they'll honk like the dickens..;)
Another possibility is the power switch itself..does it feel 'right' when you click it? it could just be stuck, or out of alignment.
Usually there's a fan connected to, or even inside the power supply..if you don't see that turning (or most newer models have one or more fans in the vicinity of the processor chip itself..the (metallic waffle thang)..sometimes.
Another thing that stumped me about 15 years ago..some power supplies have a 110/220v swtich..if that gets popped to the other side, you won't get it to turn on..
these are just some of the many possibilities..try them out, and report back to us, y'hear?..;)

I wish I'd said that.. The Staircase Wit

anahl nathrak uth vas betude doth yel dyenvey..;)