Time Machine over the network fails and reports incorrect available space
After a bit of digging into the details, I discovered that the sparse image created by Time Machine was limited to about 500GB (475Gi to be exact).  Here is abbreviated output from 'df -h' with both the network share (TMBackups6) and sparse image (disk5s2) mounted:

$ df -h
//xxx@xxx._smb._tcp.local/TMBackups6 5.5Ti 1.7Gi 5.5Ti ...
/dev/disk5s2                         475Gi 427Mi 475Gi ...


Seeing this, I double checked the "Limit each backup..." setting on the server;  the checkbox was cleared, which should mean no limits.  Still, the evidence pointed to enforcement of the default 500GB limit.

At this point I started thinking about creating the sparse image manually with the limit set to a sensible value.  However, I don't do this often a did not relish spending time figuring it out.  Instead, I set the limit to 9999GB and now Time Machine is backing up properly.  To change the limit,


Open the Server app.
Click on the Time Machine service in the left column.
Double click the backup destination on which to set the limit.
Check the Backup Limit check box.
Change the limit in the text box to be larger than the backup drive.
Click the OK button.


Here is a screen shot (cropped) showing the results of steps 4 and 5 with the check box and text box highlighted in yellow (OK button not shown):



I do not yet know if setting the limit larger than the size of the physical drive will cause problems in the future. I believe from Maverick on, Time Machine is aware of both the physical drive size and maximum size of the sparse image.  So don't expect problems.

I am satisfied with this work around for now.  If anyone knows the root cause and a real permanent fix, I would appreciate the info.