If it used to do that, and now it doesn't, it's broken.
Knowing the make and model might help with that.
Here is the tiny bit I think I used to know about RC servos:
They expect to receive a pulse every once in a while.
If the pulse is 1ms wide, they go to the center of travel.
If the pulse is a little wider, they go in one direction and stop.
If the pulse is a little narrower, they go in the other direction and stop.
The position of the output arm is proportional to the width of the last pulse received.
I think the normal full travel is +/-90 degrees, beyond which there is a mechanical stop.
Providing a mechanical stop at +/-180 degrees is possible, but adds complexity, and weight, neither of which is desirable in an RC servo.
I suppose you could make, say, a 'ten turn' servo, using a ten turn pot or equivalent for feedback internally, but again it adds complexity and weight, and I'm not aware of any such thing being sold commercially, nor of any demand for it.
I am dimly aware that people have removed the stops and disabled the position feedback function to get continuous rotation for special applications. I am not aware that any commercial servo has such an option, but I don't participate in that market.