We have purchased the cheaper Allianz Travel Policy that's offered when we purchase airfare. It has some basic trip delay/cancellation/interruption/baggage coverage and some basic medical.
The only time we made a claim, we were delayed leaving our home airport and missed our connecting flight by a matter of minutes. Had to spend the night in Phoenix. I made the claim through their website for a one night stay in a hotel. I sent copies of an email I had an airline rep send me saying that our flight was delayed causing us to miss the connecting flight and of the hotel bill. Had a refund pretty quick.