Update: so here's what happened. We all got on the plane. We taxied to the runway. The captain announced that there was a problem and we were going back to the gate to get it looked at. We taxied back to the gate. The maintenance crew came to look at the problem. The maintenance crew told the captain, who told us, that the problem was that the light that indicated that there could be a problem with the plane was not functioning properly. That's right -- they couldn't tell if there was a problem with the plane because the "hey there's a problem with the plane" light wasn't working. More of an epistemological problem than a metaphysical one, but the result was the same: we all got off the plane and waited for them to find another plane. They found another plane. We all got on the other plane and waited for them to transfer all the baggage from the first plane.
By this time it was 2 AM and we were two hours late. We arrived in Dallas ten minutes too late for me to catch my connection. Luckily there was another flight two hours later, and I did finally arrive home on the same day that I was originally scheduled. It could have been much worse.
Feel free to relate similar stories in the comments.