Swap out the RAM sticks and carefully brush any dust off them then put one back to see if that makes a difference . If it doesn't, try it in the other slot and continue trying each stick in each slot.
Make sure all the power cnnections are firmly made, especially the 12 Volt to the CPU - a small four pin connector close to the heatsink.
You could still be right about he PSU so if you have another one, try that. It could also be a simple matter of the power button failing to reach the micro swith behind it.