I've had this happen to a friend I was travelling with on an LHR-FRA flight, when she realised at the door of the terminal that she'd forgotten her passport. I sent her home in a taxi to get it and to try to get back to the airport in time for the flight, meanwhile checking myself in and explaining to the check in agents what was happening.

In the end she missed the flight by about 10 minutes, but the airline (LH) simply rebooked her on the first flight out the following morning with no fuss, no hassle and no stress. Obviously they had space, but the ticket conditions were standard European travel-or-bust (no changes for any reasons, no refunds of any amount) which would not normally have permitted this. These conditions were just waived because the airline was obviously satisfied that this was a genuine error.

I wouldn't expect such treatment. But I am always glad if I am able to choose an airline which is prepared to try to distinguish between a person with a real and unforeseen problem who should be helped, and one who is just trying it on.