How have you set up your server?
Each user should be launching filemaker and then using open Remote... to find and access the file hosted by the server. That "path" thus should be identical--don't see how it could differ from one user to the next.
"Our network administrator noticed that in his "server view" some of the databases appeared more than once, though each has only one file."
That line's got me scratching my head in confusion. What do you mean by "server view"? Is he using the Server Admin application to look at the list of files open on the server or something else? If that's what your administrator is doing, it suggests that you have more than one copy of the file open and hosted by your server and that would definitely be recipe for missing data.
Yes, our network administrator was using the Server Admin application. It seems that at some point, he moved the database files from their original directory, to a sub-folder that he created. He has now moved the files back into the main directory and deleted the sub-folder. Now, only one occurrence of each database appears as being hosted by our server.
Based on this information, can I be reasonably sure that we have remedied the problem?
Well we don't really know for sure how the data went missing! That'd require hopping in a time machine and secretely watch everything take place all over again. ;-)
You've certainly eliminated the most likely cause of the issue.
If the problem doesn't recur, I'd say you've fixed it.