I believe Delta's policy is that if you are hospitalized (or a member of your party is), then non-refundable tickets can be refunded. You need to provide the person's name, hospital name, doctor's name, dates of hospitalization, etc.

I did this last year when a party of four of us were going to travel (mostly on separate tickets) to a vacation together, and one member was indeed hospitalized so we cancelled. My ticket was booked with cash. I called up, initially hoping to just get the taxes refunded, learned of the policy, and got a full refund.
Beforehand, the person who was hospitalized had already cancelled his ticket, which was a mileage ticket, and paid the $150 change fee, as he is not Platinum. When I learned about the hospitalization waiver policy, we called back, and Delta was happy to refund the $150. This is very pro-consumer.
[Note: Delta did not ask us to sign a HIPAA waiver, so Delta was not trying, at least in this instance, to verify the information, but we were prepared to do so. I suppose if someone were to use the policy sufficiently, Delta would want to verify the information.]