Last missed flight:

We (family of 4) missed our LHR-IAH on New Year's Eve by 2 hours! My husband had to go back to our hotel to retrieve my wedding band/ring that I had forgotten and left behind. BA agent said there were no more seats on the next available flight (which was the next day--on New Year's Day) and the earliest possible direct to IAH was two days later. My five year old busted out in tears, bawling that he had to get home, so the agent took pity on us and booked us on the very next (supposedly full) New Year's Day flight--I guess someone else hadn't made their flight.
Thank goodness we were able to book a room at a decent hotel at a decent price--even last minute.

Lesson learned: When traveling with kids on New Year's Eve in London, give yourself some time.

Last almost missed flight: MIA-IAH (just last week in fact-during that horrible storm in Houston)
They changed the departure gate last minute and had not changed it on the monitors when we last checked, which was about 45 min. before takeoff--we missed the several announcements somehow (that's what having small kids will do to you). After noticing that no one was sitting near us waiting to board at the gate even though there was only 30 min. left before takeoff--I checked at a nearby gate and what do you know--they were boarding our flight! We actually were the last ones to board on that one. At least we made it!