I have an old computer (2005) which was working fine until I shelved it for a month to assemble my new one. Today I tried booting the old one up. The lights turn on and the fans (including the GPU fan) start up, but with both monitors I tried it with, they say "No Input Signal", both with HD-15 and DVI-I connectors and multiple cords. Since this can't be the fault of the monitors, I'm assuming that during the month of atrophy something's gone wrong with the video card.
Update: Turns out it's neither the monitor nor the video card; the replacement has the same problem. I'm assuming at this point there's a problem with either the power supply or motherboard, but I'm not sure how to test which, or whether it's worth continuing.