Ah. Issue fixed for me. It was all a simple matter of file permissions – although it took some time to track down!
I had put the .mo file right, but as I had to make the /wp-includes/languages directory myself, I got the wrong permissions for it (read: didn’t check it).
The /languages directory has to be world accessible, ie. drwxr-xr-x or 755. I also think everybody must be able to read your .mo-file, ie. rw-r–r– (that may perhaps not neccessary, but I consider it a safe choice).
Non-technical: Everybody must be able to go into the language directory and read the language file. Two short chmod ftp commands fixed it.
Thanks for your patience, and for bearing with my stoopid questions. I’m happy now. All that remains now, is to get my favourite theme and plugin in my own language. ??