If you can get your hands on a 2 pin wired connector to plug to the motherboard, you could bypass the case power switch temporarily. I would suspect that you could get one at an electronics supply, but I have never bought one (I have parts like that gutted from old computers) so I'm not sure what it is called. All you need to do is momentarily touch the wires to power it up. If doing that works reliably, then its probably a bad case power switch. If it has problems too, then I would suspect either a bad connection to the motherboard somewhere (like in the 20/24 pin power connector), or damage to the motherboard. The power switch on the front of the case doesn't actually turn the computer on or off. The motherboard must constantly receive a little bit of power from the power supply, even when the computer is off. The computer actually sends a signal to the power supply, telling to fully power up, thus turning on the computer. You can actually just momentarily short the two pins on the motherboard with a screw driver if you are careful to power it up. If you could find a wired connector it might be easier though (and no risk of shorting something else with the screw driver). Look at the manual
http://www.gigabyte.com.tw/Support/M...ProductID=1939
to see which pins are the power on pins.

If it won't reliably turn on when you bypass the power switch, then either the motherboard isn't reliably getting power from the power supply (bad contact from pin 9 here or its ground)
http://wiki.xtronics.com/index.php/ATX_Pinout
or the motherboard is damaged (you have already ruled out the power supply as two different units are doing it). Is your motherboard still under warranty?