I picked up a TI Travelmate 4000E for free and it worked perfectly using my AC adapter(genuine TI one from another 4000E). The backlight fires up full brightness for about a second, then dies. Inverter whines quite audibly while on, then stays quiet. LCD still works even with the BL out and using a bright flashlight shined into the display, the machine is usable.
So far, I can rule out both the CCFL tube and the LCD itself. I'm thinking maybe a recap of the inverter board, but that's just my best educated guess.