I wonder if it is the button on the front not making good contact.
First, unplug the computer and drain all static.
The button on the front has 2 wires going to a spot on the motherboard where it will short the contact between two pins. If you have the correct information for your motherboard, you can figure out which two pins need to be shorted.
Attach 2 new wires to those pins (don't solder, use push-on connectors). Make sure you have not disturbed anything and set it all back up properly. If you are then able to turn on the computer, then that button on the front is the culprit. (let it go through a few cycles, to be certain)
Other than that, most power supplies require it reach a certain voltage within a certain amount of time (about 500 ms). If that voltage is not reached, it will not turn on the power to the motherboard.
The worst type of problems are the ones that are intermittent.
I hope this helps.