I've tried re-installing Office - twice. It's an enormous pain in the tush and doesn't solve the problem. However, there is one new development.
I noticed in a post elsewhere that this seems to be either caused or exacerbated when you have multiple accounts and set the 'send / receive' settings for short intervals. It was suggested that it had something to do with the fact that if the server was running slowly or whatever, one 'send & receive' might overlap with the next and cause an error message to pop up; and because the system couldn't identify the error, it would pick the nearest available error message, which is the 'enter network password' popup. I have nowhere near enough technical knowledge to know whether or not this would be true, although on the face of it, it does appear to make sense, and indeed to explain a few things. For example, the randomness of the occurrences : the server will run faster at some times than others, there will not necessarily be any pattern to this, and therefore there will not necessarily be any pattern to the incidence of the popups. The fact that ticking the 'remember password' box doesn't solve the problem - since remembering the password was not the problem in the first place, inasmuch as the dumb server just threw up the nearest available error message it could find, clearly remembering the password will not make any difference. The fact that no amount of changing settings ( apart from the regularity of sending and receiving ) will fix the problem.
Having said all of that : my settings were originally set to send and receive once a minute. When I read the post I'm referring to, I changed the settings to once every two minutes, then once every three minutes, then once every four minutes ( which is where it currently stands ). It's very difficult to monitor precisely how frequently the problem occurs, since it can often be the case that if I leave Outlook running whilst away from the computer ( which I usually do ), there will be an 'enter network password' popup waiting when I return, and of course I've no way of knowing how long it's been there. However, it seems to me to be the case that the frequency of the occurrences may have reduced to some degree as I've reduced the frequency of the send / receive.
And having said all of that : whether or not I'm right about the frequency of the occurrences having diminished, they certainly haven't stopped altogether. Therefore, I don't know whether I'm barking up the wrong tree or not, nor do I know what would be the optimum frequency of send & receives - every four minutes clearly isn't few enough. I'll continue to reduce the frequency and see if it seems to make any difference, although in the meantime I'd appreciate any thoughts that anyone may have on this.
By the by, I have to say that even if this does ultimately provide a solution, it still seems bizarre to me that I and another multitude of users have had to come by this as a result of hideously lengthy trial and error. Shouldn't it be the case that either Microsoft have identified and solved the problem by now, or alternatively that the solution should be clearly, readily and publicly available?