Air Force Secretary Deborah Lee James became “violently ill” during a ride with the storied Thunderbirds aerial acrobatic team this week. Commanders at Nellis Air Force Base, Nev., thought it would be good PR to invite the relatively new secretary for a flight with its most famous squadron. Amid tight defense spending, there is pressure to demonstrate the worth of the Thunderbirds, which is back in the air show business after having been grounded last year when sequestration took hold. But Tuesday’s F-16 flight did not go as planned in the cockpit for Ms. James, a former House defense staffer,...