  We point out that 5D large radius doubly spinning black rings with rotation
along S^1 and S^2 are afflicted by a robust instability. It is triggered by
superradiant bound state modes. The Kaluza-Klein momentum of the mode along the
ring is responsible for the bound state. This kind of instability in black
strings and branes was first suggested by Marolf and Palmer and studied in
detail by Cardoso, Lemos and Yoshida. We find the frequency spectrum and
timescale of this instability in the black ring background, and show that it is
active for large radius rings with large rotation along S^2. We identify the
endpoint of the instability and argue that it provides a dynamical mechanism
that introduces an upper bound in the rotation of the black ring. To estimate
the upper bound, we use the recent black ring model of Hovdebo and Myers, with
a minor extension to accommodate an extra small angular momentum. This
dynamical bound can be smaller than the Kerr-like bound imposed by regularity
at the horizon. Recently, the existence of higher dimensional black rings is
being conjectured. They will be stable against this mechanism.
