Oh Noes

[Malcolm_Reynalds]

Im thinking it may be a case problem. Or the power button is just not working.
Lights in the back are on so power is still getting through.

First off you should clarify "shuts down", did your computer go through an actual shutdown, or did it simply turn of (like you flipped the power supply switch, or unplugged it)

I'd start to suspect your Power supply first.

Since modern Power supplies all ways provide some power to your motherboard to maintain a sort of "standby" state and wait for a signal to turn on. Power "getting through" is not a indication that the power supply is still good. Motherboards have a power connection known as "power good", if the power supply is outside is normal operating voltage, your motherboard won't power on when you tell it to.

Because your mention of "random restarts" and now not wanting to turn on at all I'd also look at your motherboard, inspect your motherboard and all of the capacitors on it (the small cylinders) especially near your CPU. Check to make sure nothing looks like it overheated and your capacitors are not bulging or leaking (they should have a relatively flat top flat top).

if your motherboard checks out fine I'd lean toward Power supply, unless your PC was built by HP (those power switches aren't the best)