This may happen due to overcharged capacitors. This causes the PC not to boot. To test this, power on your PC and play with it for a bit, then turn it off. Try to turn it on, and it won't work as you said. When you unplug it, click the power button once or twice then plug it back in. This should discharge the capacitor, allowing you to boot the computer once again.
Overcharged capacitors are likely due to the power supply being stressed in running all of that hardware.