I'm pretty sure this is not a permissions issue. (I have checked permissions - they seem fine.)
It occurs when I try to move a page to a new parent.
I think it is because WB is trying to do things in the /pages directory, but I don't use a /pages directory.
If I copy the directory hierarchy that I'm trying to change into /pages, and change "Pages Directory:" in Advanced Settings (to /pages), it will successfully move the page.
So it looks to me as though some part of WB is ignoring the fact that my "Pages Directory" setting is left blank, and is expecting to find the things it needs to move in /pages rather than in /.
It is generating the insufficient privileges error (I think) because it is trying to move a directory to a path that doesn't exist.
Thanks for your replies, here's some more information:
I'm using version 2.6.5 - I've been running this site on WB for years. Last night I tried to move a page, and that wouldn't work (it wasn't a problem before).
By "move a page", I mean assign it to a new parent.
Since, in Advanced Settings, "Pages Directory" is not set to "/pages", I don't know why I should get an error message referring to it.
It made me think that WB was trying to create or modify the directory structure within it.
To test that idea, I temporarily copied (at the shell command line) a section of the directory structure from / to /pages, and I changed the Advanced Settings for "Pages Directory" to "/pages".
After doing that, I was able to reset a page's parent for that section of the site without errors. This was just for the purposes of testing.
So it seems to me that when WB needs to write to the directory structure for pages, it's doing it in the wrong place.
I haven't had a a chance to examine the MySQL tables, but I will do that later to see if I can see any clues as to why.