If your T20 turns on the power LED (the one that looks like a "Z" turned slightly sideways) but there's no beeps and the LCD does not light up, then I would do the following and see if it helps with booting:
- re-seat the memory modules
- re-seat the CPU
- remove or replace the MiniPCI modem/ethernet card
If, on the other hand, the hard drive LED briefly flashes once when pressing the power button (and no other signs of life is seen), then your system has what's known as the "Blink of Death". It's a motherboard problem that seems to be related to one of the power controller ICs. I've had some luck with getting such a system to eventually boot up if all sources of power is removed for several days (pull battery, disconnect AC adapter). The usual advice for this type of problem is to replace the motherboard.
We had an IBM tech tell us once at work to do what he called a "static release". I thought he was blowing smoke, but it did work a few times. Basically, you pull the power and battery, then hold the power button for ten seconds. Then, put the battery and power back in and boot it up.