I discovered something interesting at 2am this morning. It appears that FM Server tries to re-authenticate the license certificate on the same day each month at midnight. I found this out by receiving a phone call from a frantic client who's business had been shut down because no-one could access their database files on the server. Their license doesn't expire until Sept. 2019.
From what I've been able to determine so far, when I set up their server, I had used my own separate login (provided by their IT people) to the computer, as opposed to the default Administrator login, and saved the license certificate file to the desktop (my version of the desktop). On October 7th, I imported the certificate as per the standard procedure and it all worked fine, for a whole month. Then at midnight last night, FM server decided it needed to re-validate the license certificate and couldn't find it, so it just shut down the server. It was an easy fix (even at 2am). I just had to re-import the certificate. Now I have to work out how to prevent it happening on December 7.
I haven't been able to find anything in the documentation or help system about this. I just thought I'd put this here in case anyone else comes across the same problem.