AA got me on a full flight when I absolutely had to fly due to a family emergency. I literally telephoned on my way to the airport. AA also waived all fees for me using frequent flyer miles. The FA's actually took the time and effort to ask me if I needed a ride to the hospital in my destination city.
It was one of those 'drop everything and fly' situations. I had no baggage and no personal items other than what was in my car and my purse. Before the plane took off, I used my cell phone to call American Express to book a rental car, Am Ex got me a car in a very busy tourist area.
After the dust settled I had to supply proof of the emergency to AA or all the fees would have been reinstated.
I always keep enough AA miles in my account for such a situation.