When tracking a local branch, renaming the tracked branch will not update the downstream branch.
See transcript:
avallee@gust:/tmp/repo (master)$ git co -b foo
Switched to branch 'foo'
avallee@gust:/tmp/repo (foo)$ git co -b bar --track
Branch bar set up to track local branch foo.
Switched to a new branch 'bar'
avallee@gust:/tmp/repo (bar)$ git branch foo -m baz
avallee@gust:/tmp/repo (bar)$ git co bar
Already on 'bar'
Your branch is based on 'foo', but the upstream is gone.
(use "git branch --unset-upstream" to fixup)
avallee@gust:/tmp/repo (bar)$