Flying does have some surprises, including not knowing who may be sitting next to you.
My wife and I had to make a trip from Toronto to Belfast in early 2016 because of a family emergency in N.I. Booked on BA to London's Heathrow and connecting there to BHD.
The Toronto to London 7+ hour flight, on a BA B777-200 was very full, judging by all the people in the boarding area, and we were able to get seats over the wing, for a bit less engine noise and a smoother flight. I had chosen an aisle seat with my wife in the middle seat and a lady beside her at the window.
One row in front of us, and in the center seating area's aisle seat, a very large man attempted to sit in that seat, beside a lady in the next seat. I'm only estimating his weight, but he had to be at least 375 pounds if not more. Very wide girth and he had to lift both armrests to almost fit in the seat, but still had some "overhang" on both sides. He couldn't buckle the seat belt as required on all flights, so a flight attendent brought him a seatbelt extender to see if that was enough added length to the belt. It wasn't even close, and he still couldn't get the seatbelt fastened. Most airlines will not permit the use of two seatbelt extenders, so after much trying to squeeze himself into the seat and seatbelt, as the flight crew watched, he was asked to move, and taken somewhere further towards the front of the aircraft, but we couldn't see if he was re-seated elsewhere, or asked to buy an additional ticket for a second seat, or a wider seat, or asked to de-plane before we took off.
We didn't see him when we landed at LHR early the next morning, either on the jetway when we got off, nor at the C&I and baggage claim areas, so we can't say for sure if he was allowed to stay on the flight.
Several airlines have rules (Southwest Airlines in the U.S., is one) that if a person cannot fit into a regular seat and needs more then one seatbelt extender, which isn't allowed, then he or she must purchase a second seat beside their own (if seats are available) or they will not be allowed to make the flight. Other airlines may make the person purchase a wider, with more pitch, seat in a more expensive area of the plane in order to fly that day.
Yes, some people for one reason or another, or for medical reasons can't control their added weight and it's a sad story, but airlines have to think of other passenger's safety (in case of emergencies needing a quick evacuation) and there are government rules to be followed, and not just for the comfort of other people on the flights.
So yes, it can and does sometimes happen.
@majorincident, definately go for the Premium Economy (PE) seats if you can get them. Well worth the added width and pitch, and sometimes (not always) better meals and other perks included. Just be aware that not all PE seats are the same, depending on their location in the aircraft, and bulkheads or galley or lav's nearby, plus on some aircraft (B757's for example) one row of the PE seats may not have a window, and on some planes the last row of PE seats may not recline at all, due to that row being in front of an exit row.
Best have a look at: www.seatguru.com to see how the PE seats are situated on the plane you'll be flying on. Not always totally accurate, but usually a good start before choosing any seat on your flight.