This absolutely does happen. I had this happen to me on a flight a few years ago. I was on a flight that went from DTW to BNA to MSY. I was supposed to stay on my same plane (same flight number), but because the leg from DTW was delayed, they used a different plane at a different gate to fly the BNA to MSY portion. I then missed my flight and had to stay the night in Nashville and catch a flight the next day. I of course was super irritated and raised hell that they could have 2 planes with the same flight number in the air at the same time.