The other possibility is that the backlight burned out. The T60 is old enough that it uses a CCFL (a small fluorescent light bulb or bulbs). Those tend to start burning out after about 5-7 years, which is right around the age of a T60.
If that's the problem, the good news is they can be replaced. The bad news is it's not really a do-it-yourself job unless you're very comfortable with electronics. You pretty much have to half-disassemble the laptop and screen, and there's a dangerous high voltage inverter in there which drives the CCFL (in fact sometimes the CCFL is fine and it's the inverter which has failed). So you'll probably end up paying a laptop repair shop to fix it for you.