Dark matter self interactions can leave distinctive signatures on the properties of satellite galaxies around Milky Way-like hosts. By analyzing a number of Milky Way dwarf galaxies, we were able to place new constraints on models of self-interacting dark matter which interact via a Yukawa potential. The results push the theory into a parameter space with a very specific prediction: self-interactions within satellite galaxies can be either very large (so large that new dynamical effects become important), or very small (so small that such models are usually thought of as collisionless), but not intermediate. Specifically, if self-interactions are large, some dwarfs of the Milky Way must be undergoing a process of gravothermal collapse, and this process has a number of distinct observational predictions which can be searched for in current and upcoming data.