I found a solution (for me, at least).
I dragged an old MyVideos56.db database from a backup and loaded it into SQLite Database Browser, and compared it to my current, broken, database.
Turns out that I didn't set things up exactly the same!
Under tvshow.c16 I found that the old database had paths like:
smb://SERVER1/video/TV/Buffy the Vampire Slayer/
whereas the new database had:
\\server1\video\TV\Buffy the Vampire Slayer\
Reason being that I remember that I set up the old library completely via the remote control, by going through the list of SMB servers and picking the appropriate network share. This time, I just typed in the path Microsoft-style, but forgot to include the smb:// prefix. I don't know if it's the missing smb:// or back-slashes which upsets things, but it's all okay now.
Hope this helps somebody else. I was pulling my hair out over this one because it just didn't appear to make any sense!
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The only other thing I've noticed - you MUST also have a folder.jpg file in the TV series' folder. I noticed that a couple of my TV series didn't, and as a result the banner doesn't load.