It is sharing a number of databases that are on a fileshare.
Does that mean that you have a set of files in a shared directory and your users open these files by double clicking (or using Open from the File Menu) to open these files? That is not a recommended way to share FileMaker files and can result in damaged files.
A shared database file should be opened with FileMaker Pro or FileMaker Server on the server computer and then users should access that file by using Open Remote.
All users use Open Remote to access the databases from this one Mac Computer.
What changed two weeks ago, when the problem started?
OSX update on server? Other processes added, running in the background?
Router or network changes? If it stops overnight, is the server sleeping or a separate background process running on a schedule?
"On a Fileshare" is confusing. Meaning the Mac is also sharing files? Meaning the FMServer is serving remote databases, on another computer? Or are the files on the same harddrive as the FM software? The recommendation by Filemaker is that the server is only serving filemaker. This is seldom the case.
Restarting the server with SHIFT key down to start in Safe Mode and do some harddrive repair, permissions fixes, etc is a simple, cheap mode of monthly maintenance. Running DiskWarrior every 3 to 6 months from a separate boot drive is my preferred method.