If this was my machine, I'd open the case to remove the dust bunnies, then push each card and cable gentle into it's connection and try again.
BUT repair should be approached with a manic eye on the wallet since a refurb P3 (http://www.compgeeks.com/products.asp?cat=SYS#Pentium%20III ) can be in the under 200 buck range...
We have a Hp Pavilion Pentium III running Windows ME with 256 RAM. For a while now,it would refuse to turn on. If I unplugged the power cord it would start up most times, but now it the CPU will light up but not power the monitor. If I disconnect the monitor it tells me to check for a loose cable,so the monitor is working. When my daughter had the computer it would stop responding then a few days later it would start up. It's worse now, please help!!!