LED comes on but system doesn't boot up

What great information. I'm so glad I found this forum. I have an HP Slimline PC. When I press the on button, the LED light comes on, the DVD light flickers, the fan spins like a top, the monitor light stays yellow and nothing happens. It's inconsistent. Sometimes I can press the button once and everything comes up normally and works just fine. Other times I have to turn it off and on 20 times. Sometimes I just give up.
I did a complete restore with restore disc. No change. I did the BIOS reset, no change.
I've taken the hard drive out and put it in another slimline and it works fine. I've put another power supply in and it still does the same thing. I've double checked all the cables and plugs ins. The power cable, mouse & keyboard, & monitor work fine on another computer. I hope someone can lead me in the right direction to resolve the problem. Thanks