Well, my library does not have any "partner libraries", and in fact, I pay for a library card to a library in a neighboring city, because my city doesn't have its own library.
But I've lived places that had inter-library book loans before, and there were still cases where none of the libraries in the system had a book.
If there’s something you really want/need to get, I would strongly encourage asking a librarian for help (assuming you haven’t already).
I had assumed this would be an imposition and I could handle searching myself. However, good librarians can work magic, whether it’s finding the same material in another format, arranging a loan, or even buying it outright.
The ones I know also assure me this is one of the fun parts of their jobs!
But I've lived places that had inter-library book loans before, and there were still cases where none of the libraries in the system had a book.