When it comes about refundable tickets, American Airlines refund policy states that travelers can get a refund from American Airlines online or by contacting Reservations. In case you bought a non-refundable ticket, there are still some things you can do, provided you have relevant documentation to support your case. You can talk to a live person at American Airlines reservations on one of these Airlines refunds phone numbers 800-433-7300 or 802-278-0759.