Not BA but happened to both my wife and I last week after checking in with AA priority desk. Presented the check in agent with ID'S for both myself and the Mrs. Agent tagged our bags and presented us with 2 boarding passes. I only happened to glance at the paper passes and saw my boarding group number was inaccurate as we made our way to security.. I then noticed that neither ticket had TSA precheck designations. It was then that I noticed the passenger names were incorrect.

My wife's last name is hyphenated. One of the tickets we were issued had the first part of her last name the same, but neither the first or last names on either of our tickets were similar other than that. Luckily the bags hadn't made it off the belt behind the desk yet and the agent was able to sort out the error.