The NAS backup is a Share (a separate partition) on a Drobo and the Mini is the only computer that backs up to that Share. When I set the share up, Drobo asked me if it should be formatted for Time Machine, and I did that. I have two other Macs that are backed up to two different Shares, set up the same way on the same Drobo, and those backups work perfectly. I don't know how to tell whether the Drobo sets up the Shares as Mac OS (extended ....) or APFS, but the fact that my other Macs back up fine to the Drobo (i.e., to sparsebundle files) leads me to believe that the file format is okay.
My Drobo share for the Mini is 4000 GB and my Macintosh HD is 1 TB, which is about two-thirds full. Time Machine preferences tells me the approximate size of the backup is 620 GB. When the backup to the Drobo quits the info tells me the backup was approx 60% complete. I've gone into Time Machine and seen the old files from prior backup times, but have not tried to restore them since this problem began occurring. The Time Machine backup had been working perfectly for the Mac Mini for a few years before this problem began occurring, and I did have occasion to restore from it at that time, which worked perfectly.
BTW, when I get the Backup Not Completed message, it usually occurs after a long period of a message something like "looking for space" ... sorry I didn't write that down.
Regarding the TM backup to the USB connected drive (2 TB capacity), when I erased it before trying to use it for Time Machine, I tried to set it up as Mac OS (extended ...), but now when I do a Get Info on it, it shows as APFS (case sensitive). I don't know why.
Since I first posted, I have started backing up to two disks, the 2 TB USB drive and the 4 TB Drobo Share and the backup to the Drobo Share just completed successfully, to a 1.21 TB sparsebundle. There was no looking for space message and no error info.
Also, I just tried restoring a file from the backup and it restored properly.
I am now trying to do the Verify ... when I initiate that by clicking on Verify while holding down the Option key, it seems to start, but no window pops up and nothing happens. How will I know if the verify is successful?
Is there anything else you need to know that I haven't told you?