Technically, you haven't said ... so we have to guess ... whether your bulbs are arranged in a series circuit or a parallel circuit.
I've read your question, then read it again, then read it between the lines, then read it between the words, then read it between the letters. I THINK that when you say they're connected "one after the other", you're describing all three bulbs connected in one line. There's no place in the circuit where the path splits, no 'fork in the road', and no place where the current has to decide which turn to take.
-- That's called a "series circuit".
-- Since there's only one path for current to take, all the way around the circuit from one battery terminal to the other one, every electron has to flow through every bulb.
-- If a gap opens up anywhere in the circuit, then there's no way for current to make the trip, and everything stops.
-- If one bulb lights up, then all the bulbs light up.
-- And if one bulb burns out or gets stolen, then all the bulbs go out.