I've done a few laptop and desktop repairs and I found out that the power supply units (PSU) for the desktop computers have built in safety features that will actually stop the flow of electricity to the motherboard if a bridge is detected (basically a short in the board). Usually when your board tries to power up and you see the CPU fan spin for a few seconds and then shuts off; that usually means theres a short in the board somewhere.
Common bridges from older computers (past 5 years) include worn out capacitors or transisters (buldged shape or oxidized brown residue around problem areas are common signs). Solder can also cause a bridge; you see, over time through many cycles of heating and cooling, the solder joint can actually go bad (many people call it a "cold solder joint").
SOLUTION:
If you want to spend the time and energy locating the short in your board they do sell socket testers which can determine if all your IDE, RAM, video, and ethernet sockets are good or bad. You can also buy a digital continuity tester which you can place over each solder joint to determine if there is a bridge between any two points (otherwise known as an electrical short). Once located use a desoldering braid and desoldering sucker in conjunction with flux paste to remove the old solder. Then place brand new 60/40 solder into the joint.
Due to the large number of points to be tested you'll probably not want to undertake this task, instead there are companies that specialize in motherboard level repairs. Simply google "board level repairs" and you should come up with some companies that you can ship your computer too, generally they don't charge to diagnose your computer and you'll only have to pay if they are successful in fixing your mobo.
HOPE THIS HELPS