Speaking from experience, although only one particular case. My father-in-law was planning to visit us in UK from Georgia, flying with Turkish Airline. About a week before the planned trip, he was admitted to hospital in relation to a previously undiagnosed heart problem - and he was actually still in hospital on the day of the planned travel.
The ticket was a non-refundable, non-exchangeable, etc, etc. (i.e. the cheapest possible). With a note from the hospital, the airline agreed to refund the actual price of the ticket, but not the taxes and airport fees, as they don't control those. In our case, it was about half the ticket price. The remainder of the costs (i.e. taxes, fees, etc.) were covered by his travel insurance (minus USD $25 deductible).
All in all, we lost about $25 out of about $400 cost of the ticket - better than losing everything.