We have a custom permission group that can Add Items but not Edit
Items. A user opens a document from Shared Document library (as
Read-Only.)
They make changes to the document.
They save the document with a new filename.
They are told that they don't have permission to save the file. They
click "OK".
The file is in the document library.

Any idea why we're getting that error message? What we can do to
eliminate it? (Without giving them Edit permissions.)