OK, I read the pinned post and that didn't help so, here goes:

I acquired a CPU from a local retailer, it was the display model and had been placed into the back room after they changed their setup. While it was in the back room, someone had removed the side panel and the RAM module. But, I figured for $20 it was worth it even if all I could salvage was the case. It came with no cords, no manual, no anything except the CPU. I brought it home and didn't really deal with it until a few weeks ago when I had the time. I installed a DDR RAM module (128 MB), plugged a cord in (got plenty of those), hooked up the keyboard, monitor and mouse, and pushed the button. The fans spun for about 3 cycles, like it was starting up and then nothing. Fans stopped and the monitor never flickered even once. I replaced the PSU and the problem remains, it is like something is internally shutting it down as soon as it is turned on. Now, being reasonably intelligent in the ways of Electronics, I would normally assume this points to a "dead short" somewhere but, my knowledge of power distribution within a PC is not as good as within an FM Radio. I would love to be able to build from the basic's of this computer as the Motherboard is not that bad (I researched the specs). I am hoping that this problem doesn't point to a bad Motherboard. Can anyone tell me where to look to find the base problem? The "Computer Repairmen" in my area are crooks, they sold my friend a rebuilt laptop for $400 that had an intermittent problem with the display and refused to repair it within the warranty period and then wanted to charge her $200 afterward.

In my experience that behavior is generally caused by a blown motherboard, bad ram, or a bad processor. Can't narrow it down much but if it was ram you'd probably get beep codes when you turned it on and the fan would still spin. If I had to pick one, I'd say a failed processor.

OK, I know the RAM is good because it is currently working quite well in my other desktop. So that leaves the Processor and the MB. My options are:
1. Buy a processor (about $13 at geeks.com) and hope for the best. If this fails I will know it is the MB and will have to get one of those for around $50. But, I will be sure that the problem is the MB.
2. Replace the MB with a combo (about $60) and still not know which component caused the problem.

Decisions, decisions. Is there a possibility of it being any other component? Not that I mind spending the money for a new MB/Processor but, I would really hate it if I did and the problem was still there.

No, I can't think of any other component that would cause that problem. You've already done a good job of troubleshooting out possibilities, such as a bad power supply and the ram.
So I think it is highly likely that it's either the processor or the motherboard but failures in either could cause that problem so I can't give you a definite answer.

OK, I have a MB and processor on their way, plus the thermal grease and anti-static strap (Mama didn't raise no fools here), a new DDR2 RAM module plus a new PSU and case fan for my other Desktop that I have been meaning to get. I guess the bottom line is, if the MB isn't bad, I can always order a second processor and drop it and the new MB into my other Desktop that is in DIRE need of a major upgrade. Either way, I will update you as soon as they arrive and I know what is up. Maybe I'll end up with 2 upgraded Desktops after I am through.

What did you find out about your machine? -- I just picked one up off the side of the road that does the same thing.