I'm gonna start by posting my specs
P4 2.66Ghz 533Mhz FSB :S
Soyo SY P4X400 Dragon Platinum Ultra
2x512MB Corsair XMS3200 C2PT
Radeon 9700Pro
Creative Audigy2 Platinum
420W PSU (one that came with the case)

Ok, so sometimes, my computer will randomly shut off. When turned on again, the HDD LED on the front of the case will turn on, but the general power LED will not. The PSU is supplying power to the components since the drives open and the fans turn on. However, the video card fails to initialize. Is this a problem with the motherboard?

do you have a spare psu laying around that you could possibly test. Try changing what wires are connected to what. and make sure that all the connectors are connected in all the way. i had the same problem for a while and figured out that one of my mobo power wires wasnt in all the way.

Unfortunately I don't have a spare PSU to test. Granted that this is the 420W that came with the Case, I wouldn't be too surprised if that was the cause of the radon shutoffs. This christmas, i am getting a new videocard and case, so maybe a new PSU is also in order.

Another issue is that my video card isn't exactly secure. The AGP slot hole where the screw goes is stripped and as a result, the card moves whenever touched.

that also could be the culprit. have u tryed possibly using a bigger screw, or a screw with a more corse thread? i could just be that. give it a try and see what happens. another thing. get a volt meter or a program and see how your lines are running. Your voltages could be dropping below normal, and that could also be causing it. Mother Board Monitor 5 is a really good program that will tell ya what the lines are running like and what kind of temps you are running.
MBM5 Linkage Just click on the "I Agree" down twards the bottom and the download window will pop up. Programs get REALLY close, but can be scewed by going through so much curcuitry. Volt meter is the best. But MBB5 is fine.

I have that exact same problem, nly mine happens if its either crashhed, or first thing in the morning. The video card just wont kick in. Several restarts and "turn off for seconds" jobbys usually fixes it, but it hasnt happened in a while.

My area of concern would be the motherboard. Being the initial piece of hardware inbetween raw poer and initial POST display, thats your source of worry. The PSU should be much cause for alarm unless your 12v rail has been really staked up - how much have you got on it? Im on a tagan 480W now, and my problem still arises once in a while.

screwy stuff. try pullin the graphics card out then putting it back in as securly as you can. heres something that just came to mind. The stripped out hole. take a ziptie and ziptie it through the hole REALLY good if you can find a way.

Have u tried diffrent cards? mabey your Cpu is overheating... but first yeah try the psu...psu's that comes with cases really suck because the rails are not the voltage they should be...