United Airlines has suffered any number of public relations fiascos over the past six years. But few if any match the incident Sunday afternoon on UA flight No. 3411, operating from Chicago’s O’Hare International Airport to Standiford Field in Louisville, Kentucky.

United personnel told passengers they needed four people to exit the packed plane to make room for four crew members who had to get to Louisville. (Initial reports indicated the flight was overbooked.)

When no one volunteered to be bumped,…