As AA EXP, I have many reasons to fly AA (or other carriers on AA-issued tickets. My own experience include:
1) Waking up in my home and realizing that my flight is in 30 min. A quick call AA put 3 of us on the next flight (same day standby - confirmed change). GA told us that they were waiting for us on the original flight and made several announcements.
2) Staring driving to OGG (Maui) from Lahaina 3 hrs before the flight and then finding out that there is brush fire and smoke got into a tunnel. A local guy told us that the last time this happened the road was closed for 3 days. Again, called AA and they told us to contact them once we get to the airport. Then I realized that there is another partially unpaved one-way route to get to OGG. We were lucky to get on that route before it was closed by police. We arrived late for the original flight, but AA was fine and our upgrades cleared because a lot of people missed that flight.

Either way, calling an airline before you miss the flight is always a good idea.