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This is very interesting... you have just described an identical problem that I have had with a 600E for the last two years or so. You turn the laptop on and the backlight does not light up. You can see pixel activity if you shine a bright light on the screen. Wait 5 to 10 minutes and the backlight finally decides to turn on. If I use the Fn+F7 key sequence to cycle between the LCD and external monitor, as soon as the LCD goes off, it stays off for another 5 to 10 minutes. I have checked the cable connections, the lid switch, replaced the inverter board and the entire lid (LCD panel, cable and inverter) and still have this problem. I am now assuming that there's something on the motherboard that is causing this problem. As I don't use that system anymore I haven't spent the time trying to track down the problem.
For your case, if you can spend a few dollars and some of your time, you can verify that you have (or have not) the same problem as my 600E. Replace the inverter board first as they are fairly easy to find on eBay and are relatively cheap ($10-$15). If you have the resources and inclination, swap the lid with another T2x system. You can use IBM's detailed Hardware Maintenance Manual to disassemble your T21:
T2x HMM
Good luck!