Computer doesn't start - Peculiar problem

My home pc doesn't start after I press poweron button. Hard disk rotates, led glows, fan also starts. But CMOS beep sound doesn't come and CD drive doesn't open. By pressing the button 2-3 time (or along with reset button) it somehow starts. Resetting the PC through Start->restart it reboots without any problem. If I shutdown the PC completetly and again power-on within 30-60 sec, it starts. But If I try to start it again after few hours, it doesn't start.

Please let me know what could be the problem.
 
It would help if you filled in your PC specs in your profile.
Has it always done this? What did you do just before the problems started?
An insufficient PSU or messed up power management settings come to mind.
 
pc spec: P3, 64MB RAM, Win98
Few months back it took 20 mins time to power off after start->shutdown, then it took around 20 mins time to start after Power On. I was not bothered since it took some time to start/shutdown. But recently it didn't start at all(unless I keep holding powe-on & reset button for some time). If the PC is left idle for some time, it goes power save mode, but doesn't start even if I kove the mouse or hit keyboard(goes to same state). I re-intstalled OS but didn't solve the problem. I also put new hard disk & CD RW Drive but that I was after I noticed the problem. What power management setting I should put ?
 
If the computer doesn't turn on then it has nothing to do with Windows or any setting you can change in Windows.

A new hard drive can easily be too much for your PSU. As for power management settings, try loading the BIOS defaults in BIOS setup and see how things go.
 
CMOS Battery?

Since it's a P3, I doubt that your CMOS battery is dead already, but it's easy to check. If you don't have a multimeter, take the battery to a Radio Shack or local PC shop and ask them to check the voltage on the battery. The faulty/insufficient PSU is more likely the cause.
 
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