I have a Toshiba Portege z20t but when I tried turning it on, it had a black screen. I saw the windows sign like the Press windows and enter the date and time briefly message but then the computer just turned off. However everything else, the back lit keyboard etc. was still on. I am not sure what to do.
Can you hear the laptop fan and the drive spinning? If so, I could recommend you to connect the laptop to an external monitor to see will there be any video signal going out from the laptop. If it does go out, then there's a need for LCD inverter replacement. If there's nothing showing either on external monitor nor display, but the fans are still working, the problem that you have may be due to the failure of the video chip. Sometimes, after much of a use and less cooling the graphics processor which is responsible for video signal transmission get unsolders, the thermic paste goes away and it gets unsoldered, therefore the screen doesn't show anything. If you possess knowledge of notebooks' hardware you can change the paste yourself by opening the laptop and check if the chipset that is next to the processor has got unsoldered or you can bring your laptop to a technician to get it soldered again.