Thanks - I tried that. The PC just reports incorrect password. After disconnecting the iPad, it then connects with no further problem.
I wondered if there is a setting somewhere in the individual user's mail configuration settings to "allow multiple connections" or whatever. I've not been able to find one, and the user (not a technical person by any stretch of the imagination) doesn't remember anything like that. The "helpdesk", of course, can't help.
I suspect, from a lot of Googling, that there is in reality only one POP mailbox into which everything goes, and which only allows one connection at a time.
I offered her a mailbox or three on one of my domains, but she wants to be independent. She has a Gmail account, but doesn't like webmail or Imap.