I have had this happen.

After quite a few struggles at check-in, an outstation with entirely outsourced staff, they finally got a boarding pass printed. They didn't quite understand the problems. They searched by last name for bookings rather than booking reference at check-in. I boarded early, and after a bit another passenger came and said, "sorry, you are in my seat" I pulled out my boarding pass for the the same seat, and we shared first and last name. Since it was an absolutely full flight, I was kind of worried. Ground staff came, took my boarding pass, went away to check. While waiting in the galley the captain assured me that if there was no seats I could have the jump seat instead. (two flights cancelled that day due to weather) Sadly no jump seat, there were actually two people booked with the same name and there was a seat for each of us.

Not BA by the way.