I was able to cancel a non-refundable American Airlines ticket within 24 hours of making the reservation using a combination of TYP and cash. There was no charge for the cancellation when I called the TYP travel department.

Note that as of 4/1/2016 AA allows free cancellations of purchased tickets within the 24 hour window even though they continue to have a hold option on their website. Previously, they used the hold option as their only way of complying with the federal requirement of offering a 24 hour refund.