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

The easiest way to track what's wrong (requires some experience with computer building perhaps) is disconnecting all hardware and then reconnecting everything one by one.

So connect the motherboard with the power supply with only RAM and your CPU in. Does it do anything? Great, add your GPU or your hard drives. Keep adding hardware until everythings connected and it's not short-circuiting (which is probably the problem). This off course is a lot easier if you have an on-board VGA connection, since then you can also read the BIOS without having a GPU connected.

I had a similar problem once, turned out my soundcard didn't really fit with my casing, so if I screwed it to my case, it would be plugged slightly out of the motherboard. Result: didn't boot!

You are correct when you assume the PS, MB, & CPU. It could have been a power spike that killed it or just overheated or just old age. Since you don't have parts to swap, it's hard to go forward. If you get fans, the PS is probably good.

It looks like the MB/CPU might have died on you...

EDIT: Since there is no beeping and it is random, I believe that your CPU is failing on you. I had a similar situation recently. If you can replace the CPU, then you might be okay to keep using the same system.

-JFV

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.).

Might sound silly but.. are you sure that there's nothing making ground? I've had experiences of odd situations where the motherboard itself or a PCI card was grounding with the chasis and refusing to boot

I had a problem like this once, turned out to be a motherboard problem. Replacing it fixed the thing right up.