I agree that my "Jr" has never prevented me from flying, but it prevents me from doing online check-in with AA. I had AA change my last name in their system to be "[LASTNAME] JR" and that fixed it. Adding it as a suffix DID NOT work, so I removed the suffix and just changed the last name to match. Just remember that when you log in to AA.COM, your last name is now "[LASTNAME] JR"