Computer doesn't always turn on...

I built my parents an e-machine about 2 months ago, with usable components from their old computer (power supply, hard drive, DVD drive, graphics card) and with a couple new components (CPU, motherboard, RAM, case), and recently it started to not power on when I pushed the power button. But, sometimes I could unplug the power cable to the power supply, plug it back in, and the sucker would start up. The thing is, that doesn't always work. When the computer is on, it'll stay on until you decide to shut it down; it doesn't randomly power down.

So, I'm thinking the power supply is shitting the bed, or possibly the new motherboard is acting up (based on what I have read from people with a similar problem). I don't think anything is shorting out. I do plan to make sure the power switch isn't loose, since I thought of that as I was typing this   I'd still like to hear from you all, since some of you may of experienced something like this before and have a good idea as to what is causing the problem.

Thanks!