I think it is one of two things, the sessions really are hanging around, or banning is a true possible as well. You should be able to prove the banning issue by testing with 3 ids, test 1, test 2, test 3 in turn let each get blocked by the connect count in turn, then try test 1 again. If it works say with in 15 minutes, that might be a session issue, if they are all blocked for a longer period, as the administration/support what their policy is on banning, concurrent attempts, and duration for repeated attempts. Be sure to ask if they have a successful login policy as well as a failed login policy!
Many FTP admins set a frequency per time limit for even invalid logins. This is done to avoid various DoS attack scenarios. As or valid logins, they could have all kinds of limits in place.
I worked for a company that did a lot of bulk data moves via FTP, it was statistical market data that was useless to any hacker unless they knew the context, so we used FTP for moves at night, and some of the receivers kept rejecting data. Turned out they not only throttled to connections from same sources, but also limited the volume of logins in total per time segments. Someone had the great idea that no one should be moving bulk data at night! Geez.