Summary: If the polaron coupling constant \(\alpha \) is large enough, bipolarons or multi-polarons will form. When passing through the critical \(\alpha _{c }\) from above, does the radius of the system simply get arbitrarily large or does it reach a maximum and then explode? We prove that it is always the latter. We also prove the analogous statement for the Pekar-Tomasevich (PT) approximation to the energy, in which case there is a solution to the PT equation at \(\alpha _{c }\). Similarly, we show that the same phenomenon occurs for atoms, e.g., helium, at the critical value of the nuclear charge. Our proofs rely only on energy estimates, not on a detailed analysis of the Schrödinger equation, and are very general. They use the fact that the Coulomb repulsion decays like \(1/r\), while ‘uncertainty principle’ localization energies decay more rapidly, as \(1/r ^{2}\).