Possible symptoms of a bad PSU but not be the case are:
1) Computer turns on, fans spin, no post noises. (Or "Beeps" if you will). Nothing happens on screen but computer is on. This may also be caused by the CPU not in the socket properly. (Some people remove the plastic AND pin guard by mistake, you're only suppose to remove the plastic piece)
2) Computer turns on, fans spin, you hear the regular single post beep. 10 seconds later the computer suddenly shuts off, one of the lights on your Mobo turns red or remains lit. (This may also be caused by overheating, something is too hot and is shutting down)
3) Computer turns on, fans spin, you hear the regular single post beep. You boot into windows fine. Everything runs great until you start a demanding application. Once the application runs for a few minutes, your computer suddenly shuts off without warning. (Your PSU is too weak)
4) Computer turns on, fans spin, suddenly shuts off. If you power on, exact same thing happens. (This may be caused by a short. Make sure no metal contacts are touching the motherboard that they need to, like unused "gold" spacers that you screw your mobo onto the case")
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Odd ball suggestions
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1) Computer turns on, fans spin, nothing on screen and keeps running. Try using another keyboard, your PS/2 port may be bad, or the keyboard is freaking out enough to make the bios confused.
2) Computer turns on, fans spin, then shuts off WITHOUT hearing the post "beep". This is a possible bad power supply, so try and see if you can use another one. Also check if you may have your PSU on a different voltage on the back (Such as 240/220 on by accident)
3) Computer turns on, fans spin, then shuts off but you hear the post "beep" and it seems like it was booting normally. Something is running way too hot, find out the problem.
After all the above and you try another PSU with the same problem, the only issue I can say is that your motherboard may be damaged or the power source you are using is so unrealiable its causing a problem. (Basically if you're using a wall outlet that is powering other things such as microwave, giant TV, air conditioner/heater, etc that not enough current is coming through to the PSU) Also, for the heck of it, see if you can use another power cord. (The black cord that goes into the back of your PSU). Sometimes that cord just goes bad.
Hang in there, good luck. I know how sick to the stomach it can be when we build a new rig with shiny, expensive parts and it doesn't work.