Computer won't turn on

For the last few weeks, my computer (Dell Dimension 4600 with WinXP, bought in mid-2004) has been odd about turning on. It works fine when it's actually running, but if I unplug the power strip it's connected to and plug it back in, everything else on the strip (monitor, speakers, etc) turns on immediately but the tower does not. Pressing the power button does nothing. For about five minutes it will just sit there, and then the indicator lights will begin blinking on and off. If I press the power button during that time, it finally does turn on.

Unfortunately I've been busy lately and have neglected to look into this problem, mostly because as I said the computer has no problems once it actually turns on. But the problem's getting worse - it takes longer and longer before it's willing to turn on, and tonight I waited fifteen minutes before coming to the other computer to type this post, thinking that one had died for good. (I checked back and it finally had started blinking, but I'm worried that one day it really will not turn on again.) I'll take any help I can get. Thanks.

Jam it back in, in the dark.

I actually called Dell, and they had me toggle the voltage switch on the back of the tower back and forth once, which cleared up the problem instantly. Thanks for all your suggestions though, guys!

There's nowhere I can't reach.