  We show that it is impossible to improve the high-energy behavior of the
tree-level four-point amplitude of a massive spin-2 particle by including the
exchange of any number of scalars and vectors in four spacetime dimensions.
This constrains possible weakly coupled ultraviolet extensions of massive
gravity, ruling out gravitational analogues of the Higgs mechanism based on
particles with spins less than two. Any tree-level ultraviolet extension that
is Lorentz invariant and unitary must involve additional massive particles with
spins greater than or equal to two, as in Kaluza-Klein theories and string
theory.
