Does anyone have an example in mind of a ring $R$ for which $R^n\cong R^m$ as $R,R$ bimodules for some positive integers $n\neq m$?
I would be a little surprised if someone showed no such thing could exist, but that would also be a welcome answer.
P.S.: Naturally such a ring could not have IBN. I don't recall deciding whether or not the "easiest" ring without IBN (the endomorphism ring of an $\aleph_0$-dimensional vector space $V$) precluded this, so that is a starting point.