Bad caps on the motherboard, it can't get enough juice to trigger the power sequence. You could probably get it going (and this is a test to determine as well) by jumping the power supply when it's connected to the motherboard. I've done this with many PC's that had bad motherboard caps and just wanted to keep the PC on all the time. Sometimes it's the voltage regulators or anything things on the MB as well.
Disconnect the power from the back on if the unit. Get a paperclip and cut it or a piece of wire that can fit between the green pin on the power supply connector going to the motherboard and a black pin next to it so a connection is made between the green and black pins. Push it in the top of the power supply connector good. Plug the power in the back of the PC and press the power button.
You PC may boot and this will help determine if it's the motherboard or your PSU.