This is not a linux issue but it is definately hardware.
Yesterday I tried to boot my system only to find out that pushing the power button resulted in a non functional system. To be specific the LEDs lighted and the fans rotated but there was no screen output (eg bios post messages). I checked and my screen was normally on. After pushing the button again the system just shut. Then I did the same things again and again but the system wouldn't boot. In a desparate way to make it work I turned off the PSU switch (at the back of my tower) for a few seconds. Then I gave power again pushed the button and ... Voila! Booted as nothing had ever happened.
The same happened this morning but...
I switched off the power. Then back on after 2-3 seconds. I push the button and see the first screen (a few lines on the upper left part of the screen about the nvidia agp bios or something like that) normally. Then the second screen about AMI BIOS. Now this is weird. I got the first 5 lines or so NORMALLY. I also saw the red AMI logo. And then the rest of the lines appeared but not normally. They were fading off and then the system sort off died again. The fans were on and the LEDs too but nothing else. The screen was blank. I noticed that there was no hard disk noise which means that there was no OS booting for several seconds. I pushed the button and the system went off immediately. Then I cut the power supply again for many seconds (about 30) and when it was back on the system booted perfectly.
So the question is:
1. What's going on? Is it a PSU problem?
2. Can this damage my hardware?

I hope it is not serious.

Same happened with my machine a couple of times.

I stopped pulling the monitor's power off the PSU and instead hooked the monitor into the power mains.

That Solved it for me.

Hope this helps

Unfortunately my monitor is already on an independent power source so this won't solve the problem. I might try my old pc's psu to determine if it is a 100% psu problem (just in case the m/b has something to do with it)

I had a problem a few weeks ago when I plugged an electric space heater into the same circuit that powered one of my computers. I found that the CPU was having symptoms as if I had overclocked it. I had apparently put too great a load on that circuit. The line voltage must have dipped below what my computer requires. My BIOS, which was made for overclocking, reset the CPU clock multiplier from 12 to 11. My /proc/cpuinfo file said that I was running at 1300 MHz instead of the correct speed of 1734 MHz. I took the space heater off of the circuit, reset the clock multiplier in BIOS to 12, and it is running properly.

Or, your problem could easily be a bad capacitor on your motherboard. Or, make sure that your PSU has more than enough power to provide more power than your computer needs. People laugh at me when I put a 450 watt Antec True Power PSU into a desktop computer but I'd rather have components that are overrated for the job than have components whose ratings barely meet the theoretical requirements of a computer. I have read many times about mysterious problems that were eventually diagnosed as having too small a PSU on the computer.

LazyP, I added a bit to this post after you read it. I originally wrote it early in the morning. When I reread it I realized that I should have included more information.

WOW. I have this pc for over a year and every now and then I get messages during bootime saying "Overclocking failed"
Of course I haven't overclocked. In fact this happens even with the defaults. Worse still that was the first thing I saw when I booted this pc. Then I would reboot and everything would be fine. This seems to be serious and I think I'll have my PSU replaced with another of bigger capacity.
BTW this is my hardware:
350W "No Name" PSU
Ath64 Clawhammer s754
Asus k8v m/b (with a via chipset which is not supposed to be energy-consuming)
2 HDs and 2DVD-ROMS
3 PCI cards
Only 1 extra fan for the case
WTF this is not so bad for a 350W PSU but maybe it was defective since I bought it.