I've gone through this more than I care to remember this year- suffix caused ticket/ passport mismatch last year on trip to Europe. Called AAdavantage customer service and explained situation and after bouncing through several departments, they finally got my profile updated.
However, they told me that the systems that talk to immigration control systems would be set to match passport, but when logging on to AAdvantage website, suffix still would not display. Upon further calls to AA for another trip, was revealed that even on internal AA systems, suffix shows up in Address field, but not in passenger profile section.
In any event, always lots of anxiety and consternation when trying to check in online for international trip and get " cannot check in electronically, please see agent at airport" message. I THINK ( will find out soon ) that problem to/ from Europe was caused by bona fide PNR/passport mismatch due to suffix. Trip to Hong Kong couple months later returned same error message on electronic check- in attempt, but after several phone calls to AA, was informed that trips to certain countries, including China, ALWAYS require in person check-in.
After updating name profile with AMEX corp travel dept to avoid issue with international travel, found that apparently they do not have a separate field for suffix, so they place suffix in "last name" field. Now, anytime company travel is booked through Amex Travel, I have to call airline and have them manually reconcile PNR to match AAdvantage number. So now, I have huge PITA with business travel domestically in addition to aforementioned joy.