3 or 4 years ago I was headed to STX from DFW with a connection in MIA. STX flight was the last of the day, and it was Thanksgiving week. Connection was a little tight and I worried about it. Finally called AA and persuaded some AAgent to move me to an earlier outbound flight even though that gave me 4 hours and 10 minutes in MIA - technically a stopover, which was not allowed on my fare. I made the connection just fine, settled into my seat for MIA-STX, and they closed the doors. We sat at the gate for a while, and they finally reopened the doors for a half dozen or so passengers who had very obviously rushed for the flight. One of them had the seat across the aisle from me and I finally asked where they had come from. Turned out they had come in from DFW on my original flight, and AA held the MIA-STX flight for them about 45 minutes.

So it does happen, but I am still grateful to the AAgent who took the stress out of my holiday trip by bending the stopover rules by 10 minutes.