Do American Airlines Refunds tickets?

It depends on the carrier and exact ticket conditions for your fare code.
(I don't mean by class - economy, business, etc. - I'm referring to the specific code that applies to your ticket. Each class is divided into several single-letter codes, each with its own specific conditions. Some codes are normal. For example, 'Y' for the full-fare economy, 'F' for full-fare first; most are specific to other individual constraints.)American Airlines Refund
If the ticket was upgradeable, you would have been able to pull a little trick where you upgrade it to the American Airlines Refund Refunds refundable fare class, then later cancel the upgrade ticket for a full refund.