It's always possible. you'd have to measure the battery voltage, at the controller's battery connector input, to find out. You should test with the light off, and on, and see what the voltage is either way. If there is a large voltage difference, then there is either a problem with the battery itself, or such a bad connection*** somewhere between the cells and the controller that it is causing that voltage drop under load.
***bad connection can mean corrosion in a connector or wire, broken wire barely touching something, connector not fully plugged in, etc.
More details about the actual problem you see, actions taken to test, what specifcally happened just beofre the problem started, etc., may help us help you. More info is almost always better.

(quite often, the information left out points directly to the cause of the problem and thus directly to a solution).
If the whole scooter (not just the throttle as stated in the OP) works normally without the light on, that eliminates the battery itself as the light shouldn't take nearly as much current as the battery does.
If the scooter doesn't operate normally but just the throttle does, then it could still be the battery (or connection).
If the lights are correctly operating when the scooter does not, another possibility is the switch itself, if it has multiple positions like a car ignition. If there is a position for "just lights", and one for "just motor" and one for "lights and motor", and the switch is definitely in the right position, then it could be broken internally so the positions are off by one, etc.
If the lights are not operating, and the scooter doesn't operate, at the same time, when the lights are turned on, then it's much more likely to be the battery (or connection) being overloaded by a short circuit (wiring, electronics, etc) in the lighting circuit past the switch for the lights.
If the lights are controlled by a button on a cotnroller screen, then the screen itself (or the controller) could have internally failed so that switching the lights on forces a shutdown of the controller system, short circuit, etc--this event would almost certainly prevent the lights from operating as well.