I have personal experience here... last christmas (13 months ago) returning from UK to USA, first myself, then my wife, came down with some food borne illness the night before travel... we were both up during the night vomiting.
We phoned AA (in the US) in the early hours but they declared that all alternate flights were full for the next couple of days. I gave up and phoned AA in the UK (when they opened at 7am)-- only a few hours before our original departure time. They could not have been more helpful and after some time they asked whether I could provide a Doctor's note. I said yes and they re-accomodated us on a flight the next day -- then I threw in the zinger... would my VIPOWs still get us (family of 4) in business... on hold again for a while then they came back with afirmative, on a MAN-ORD routing (I was originally a LGW-DFW). They even rebooked our BA connection.
As for the doctor note... my father is a retired physician so he wrote a note on official stationary declaring us unfit to travel and faxed it to AA in dublin as requested. No questions asked.
I should repeat that on this occasion AA could not have been more helpful or accomodating. They treated us extraordinarily well both on the telephone, and when we arrived in MAN with LGW-DFW-xxx tickets needing reissued as MAN-ORD-xxx tickets (they sent us to the BA lounge and then showed up 45 minutes later with new tickets and boarding cards).
DAK