How to troubleshoot a computer not booting up when powered on?

"There are lots of things that would keep a system from always powering up. It would take an experienced engineer a few minutes to days to debug an issue you are describing. As mentioned in other Answers, it could be the GND, one of your add-in cards or modules or your motherboard components that have a problem.

Some other things that I can think of from my experience/perspective (although improper assembly and/or bad add-in cards and modules are much more likely to be the issue):

A incorrect BIOS setting (try to update BIOS, clear the CMOS and use default or safe settings).
DC-DC power supplies may have design issues which keep them from always powering up.
The power up timing is out of whack or power sequencing control wasn't thoroughly validated.
The actual important components on your motherboard may have an issue which had gone unnoticed or the component manufactuer made a ""business"" decision that the issue wouldn't affect anybody.
Capacitors on your motherboard could have failed, maybe you can see the electrolytic cans casing bulging outward.
AC silver box (ATX Power Supply) didn't reach it's minimum load so that it can power up. Some power supplies require a certain load on their different power rails before they will power on.

(I'll add more to this list if I think of any.)

Generally debugging a board requires taking it out of the chassis, getting a hold of the schematic and layout design files, and taking measurements with a meter and oscilloscope to determine the exact cause of the failure after the easy stuff is taken care of (testing with a known good configuration, clear cmos, etc.)."

"Like others here, my gut instinct is pointing at the CPU.

I had a situation like this a few years back, and it turned out that the CPU and MB were both fine, but the cheap thermal paste connecting the heat-sink to the cpu had broken down to the point where it wasn't making a firm connection between the two anymore.

I was ready to replace the CPU, but on a whim I picked up a new tube of thermal paste for 5 bucks, applied it, and the machine has been running perfectly ever since."

Does it beep when it fails to boot? It could be failing to POST. If there's a beeping pattern, you can check your mobo manual (or website) for what the pattern of beeps might mean.

Hardware problems like this are very difficult to troubleshoot, but I have personally run into the exact symptoms you describe on a couple of machines and it turned out to be a bad power supply both times. In each instance, the fans spun up and the lights came on as you mentioned, but the machines would not POST / boot. The power supplies even tested OK according to a power supply tester tool, but both times, swapping in a known good power supply fixed the issue straightaway.