One problem you might find is as the tubes get older they need a higher start up voltage and more current to get going. And one of the controller chip families, an example is OZ964, is intelligent in having open lamp and over voltage protection, so it may decide the lamps are faulty and shut down, even when they are still usable. I have a Dell 2410 that does this, the lamps flash once and that's it.
You could replace all the CCFL's in the hope of coaxing it to work again, or try modifying the controller turn off parameters, however another option is to replace the lot with LED strips and an appropriate controller. This has a lot of advantages, cheap, no high voltage, and it wont shut down if a LED goes out. You would need to use a fair number of LED strips to get the same quality of light output, however LED's shine forward so all the light produced is going in the right direction.
I used to use a Tosh backlight inverter as a quick method of driving CCFL's, just wire the shut down control on.
Ken