|If you get no display but the computer seems to be working otherwise....|
Try plugging a monitor into your VGA port, when the computer is shut down, then booting to see if the display is then normal on the external monitor.
See response 1 in this:
I found a review of thinkpad t40 cpm dated October 2006, so your laptop may be that old or older - your laptop probably has been used a lot.
"When I was using it, the whole laptop froze and nothing worked so i turned it off, and now it won't turn on."
If your cpu is overheating, your laptop will work fine for a while after it has been booted after having had a chance to cool to room temerature, but when the cpu has gotten too hot, the laptop will mis-behave, and most more recent mboard's bioses will shut down the computer eventually, and in that case the computer will not start up again until the cpu has cooled to below a specific temperature.
Make sure there are no obstructions that prevent air from getting into or out of openings in the base of the laptop. If that doesn't help, there may be too much mung (lint, dust, etc.) inside the laptop in the air passages, or on the cpu heatsink, or the fan that cools the laptop may have too much mung on it or be be failing or not spinning at all (laptop fans are usually temperature controlled, though they may come on for a short time when you first boot in any case) . You are wise to consult a service manual for your laptop in order to find how to properly open it up and check for those things.
You can find the current cpu temp in most bios Setups. Of course, the computer has to be working and have video in order to check that, and you should check that after the computer has been running a while.
If the laptop freezes at random times even when the cpu should be cool, lots of things can cause freezes.
Sometimes that's caused by a hardware problem other than overheating.
Make sure the main battery is properly seated and making a good connection in it's socket. E.g. unplug it, plug it back in; examine the connection on the laptop.
It may be you may have a ram connection problem
A common thing that can happen with ram, even ram that worked fine previously, is the ram has, or has developed, a poor connection in it's slot(s).
This usually happens a long time after the ram was installed, but it can happen with new ram, or after moving the computer case from one place to another, and I've had even new modules that needed to have their contacts cleaned.
See response 2 in this - try cleaning the contacts on the ram modules, and making sure the modules are properly seated:
For a laptop, you must remove both its main battery and AC adapter before you do that.
Consult your User manual for your model if you need to.
If you have not replaced it's original battery, you probably should do that.
Your main battery may be too old and/or defective, and/or there may be something wrong with the AC adapter, or the AC adapter's connection to the laptop.
See response 1 in this:
Correction for that:
The jack in the laptop should move when you wiggle the plug in the jack.
should have been
The jack in the laptop should NOT move when you wiggle the plug in the jack.
The power jack in the laptop may have a mechanical switching device that switches the laptop to main battery only use. If the jack is damaged, that feature may not work and the computer won't start up when the AC adapter is disconnected even when the main battery has a sufficient charge and there's nothing wrong with the battery. In that case you must replace the jack.