Unfortunately, POP didn't really completely solve problems for all situations. Indeed, POP actually introduced new problems which hadn't existed with the NFS solution. The most obvious problem with POP was most visible in the university environment.
The problem was that POP essentially presumes that a given user and his/her MUA was always going to be on the same machine. Once the e-mail is delivered to that machine, there's no standard way to access it from another machine. Many universities, including Carnagie-Mellon, had installed workstations throughout their campus for students to use. These machines were used in a very similar fashion to the public terminas that replaced them in the sense that they were shared by the entire student body. This meant that most students did not always use the same machine to read their e-mail. The NFS solution didn't have this problem, because all the mail was still kept on the file server, which was accessible from everywhere.
Problems with POP weren't limited to this situation however. POP also had problems with people accessing their mail remotely using modems. In particular, POP wasn't very sophisticated. A user could login, find out how many messages were on the server, and download those messages. However, it didn't provide any way to grab portions of messages, and it certainly didn't allow for summary information about messages to be downloaded. This meant that remote users frequently experienced long delays as they waited for a huge mail message that they potentially didn't really care about to be downloaded (probably some silly picture) before they could get to the other important messages which might be on the mail server. This also meant that if remote users were being sent large messages they didn't want to receive, they had to download the message before they could delete it. This problem persists even today, but keep in mind this was happening in an era before 56K modems. ;-)
POP also created problems where multiple users shared the same mailbox. A good example of this would be a help desk. When a new message arrived, everyone on duty at the help desk needed to be notified of a message and know it's status. With POP, whoever logged in first got the message and could either download the message or leave it on the server. In the latter case, noone else could tell if it was being acted on. There could also been lock contention issues.