  We derive new positivity bounds at finite momentum transfer, assuming a large
separation between the mass $m$ of the lightest particle in the effective
theory and the mass gap $M$ to new heavy states. Massive gravity parametrically
violates these bounds unless the cutoff is within one order of magnitude of the
graviton mass $M\lesssim O(10)m$. Non-gravitational effective theories of
massive spin-2 particles are similarly bounded.
