Welcome to thinkpads.com!
I can point you to this conversation that's currently running here in the T4x forum:viewtopic.php?f=2&t=76764
as that person also seems to have the same type of problem. If you reduce the brightness of the display (Fn+End), does it appear that it stays on a little bit longer? If so, then your problem may lie in a defective inverter board, an aging backlight inside the LCD panel or some other problem with the motherboard. The easiest and cheapest thing to do would be to replace the inverter board with a known working unit and see if that helps. Those boards usually run around $15 (shipped) off of eBay. In order to tell you the correct part number, you will need to supply us with the seven character TYPE number from the bottom of the system (e.g., 2374-ABC). It's also a good idea to put that TYPE number in your signature field here in the forum so that way people can see what your system configuration is and therefore provide better answers to any of your questions.
If you (or somebody you trust) are going to work on the system, then I would suggest first downloading a copy of the Hardware Maintenance Manual for the T4x systems from here:http://www-307.ibm.com/pc/support/site. ... MIGR-46464
It has all of the instructions for taking apart the system.