It seems to me that if an airline has a record of a person flying, but thinks that it is possible that the person was not on the flight, it would be a MAJOR breach of security.
They have to do a head count, they get the boarding cards, usually through an automated check in process, they have checked identification, etc. etc. etc.
If a passenger is NOT on the plane, then they have to check for bags, and remove it there were. So when they do the head count, and compare the collected/scanned boarding passes with the manifest, they should be dealing with any discrepancies, otherwise they could be letting someone from another flight onto their flight, or vice versa.
If it turns out that the airline's records are wrong, then the airline could be opening itself up to investigation, law suits, etc.
So how does one prove one was on a particular flight? I suppose you'd need receipts before/during/after to show where you were, as well as credit card receipts for the purchase of the ticket, and the boarding pass, maybe phone records to show you at the layover point, etc. But I don't think you can prove absolutely that you were on a flight, unless you can get passengers who can all vouch for each other...