I'm not sure about laptops, but sometimes when you power on the machine, there is a screen before the main BIOS screen that gives details of the display device. This screen sometimes dissappears before the display has a chance to initialise, but if it is at all visible I often reset the machine several times until I can read all of it and write it down.
If you have Windows Update that might give you a driver.
Also, remember that if the maximum you can select is 256, then you are referring to 16 colours, not 16 bit. You have 16 colors, then 256 colors,. Then you have 16 bit, 24 bit and 32 bit, which are higher settings than 256 colors.