Synology: Time Machine backup keeps stopping without errors EDIT: It appears this may not in fact be a Synology issue, as I'm now experiencing the same behaviour trying to run a Time Machine backup to a newly-formatted local USB drive.
I just bought a Synology DS220j mostly to replace my dying Airport Extreme, which I was using for Time Machine backups from my MacBook Pro (10.14.6). So far, I've been unable to complete a single TM backup to the Synology.
The main problem is that the backup always stops after backing up only a few tens or hundreds of megabytes, usually with no errors in the Time Machine preferences pane or in the Console, but very occasionally with a &quot;Could not complete backup&quot; notification.
I've tried doing a new backup from scratch, and I've tried seeding a backup from my last good sparsebundle file on my previous backup disk (after fsck-ing it for errors). I've tried over SMB, and I've tried over AFP (which I've now turned off again, as I know it's deprecated and generally doesn't play too well with Synology).
Currently, I'm running a cron job on my Mac that runs 'tmutil startbackup' every 5 minutes. This is my second attempt at that—the first time, it nearly got to the end after a couple of days, but then the &quot;amount remaining to backup&quot; just kept increasing faster than it was backing up, so that backup never finished either. (I don't think that's a Synology problem, though—seems to be a known issue when using TM with a NAS, though I don't know how to solve that either.)
Can anyone suggest anything to make this less painful, and preferably, Just Work?
New information has come to light. The Mac may need directory repair, significant space freed or just an erase install based on multiple clean Time Machine starts failing.

It’s hard to pick apart backup issues with such a custom setup.
I recommend you start with the “nurse” procedure at the bottom of my answer - working up to fsck and safe boot as appropriate. Exclude three folders and let a minimal backup complete. Consider throwing away the old backups and starting with a clean slate, no restrictions or fancy settings - let Time Machine think it can use the entire disk.
/Users
/Library
/System

If the interface asks to skip all system files, say yes. Once you have hourly backups going well. Remove things from the exclusion one by one and focus on disk corruption on the source if that’s where the backups hang.
Also, disable cron forcing things while stability is not present. Let the normal triggers work until you’re solid is my advice. Also, The Time Machine Mechanic and Backup Loupe are invaluable tools for me working on problematic Time Machine situations. Yours seems overly not easy, so something is causing you pain that hopefully can be identified and remedied.