Echoing N6, rtw "problems" almost always involve buying tickets and revising those tickets after starting - the rules are really not particularly complicated, but some are badly worded and subject to intepretation, and some important things are unsaid, so too often there are differing points of view about how much a ticket should cost or what travel should be permitted.

About the only "en-route" concerns anyone mentions concern seat availability. The rtw fares only permit the traveler to book flights that have seat availability in the appropriate discount fare category - A, D, or L for first, business, coach - and some travelers are annoyed when a flight appears to have lots of seats in their chosen cabin, but they're not allowed to book it. That leads to a little venting here, but I've never seen a "rtw trip from hell" kind of report where one hapless traveler had enough "experiences" to make a tale out of it.

And lest I've frightened the OP regarding seat availability, although some airlines are notoriously stingy about releasing first-class seats on prime routes (e.g. American on its coast-to-coast services), availability is startlingly good imo. Unfortunately, it's impossible to quantify that meaningfully without a page full of anecdotes, but if you plan a month ahead in first class you can have any flight you want (with exceptions); if you want to travel "tomorrow", it's 50-50. "Tomorrow or the next day or so" and you're usually back to 100%. (Availability in the bigger cabins becomes progressively better, I believe.)