I remember doing this once around 1990, I think
Flying CO AUS-IAH-CUN. The IAH-CUN aircraft was actually on the tarmac at AUS due to a weather delay from CUN-IAH
My flight to Houston was delayed, and there was no guarantee it would get there before the IAH-CUN flight would take off.
So, we told the GA we were renting a car in order to not miss the flight to CUN. We rented a car, and shared with 2 other passengers who were also trying to catch the flight to CUN.
The AUS-IAH ended up getting to IAH too late for that IAH-CUN connection. So, not AA, but I suspect AA would allow the same. In that case, the flight wasn't evened cancelled.
We also got refunded for the leg we drove. No problem with the rest of the segments as we had told the agent before we left.