  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 explodes? 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 Schr\"odinger 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 $1/r^2$.
