  Determining the range of validity of Migdal's approximation for
electron-phonon ($e$-ph) coupled systems is a long-standing problem. Many
attempts to answer this question employ the Holstein Hamiltonian, where the
electron density couples linearly to local lattice displacements. When these
displacements are large, however, nonlinear corrections to the interaction must
also be included, which can significantly alter the physical picture obtained
from this model. Using determinant quantum Monte Carlo and the self-consistent
Migdal approximation, we compared superconducting and charge-density-wave
correlations in the Holstein model with and without second-order nonlinear
interactions. We find a disagreement between the two cases, even for relatively
small values of the $e$-ph coupling strength, and, importantly, that this can
occur in the same parameter regions where Migdal's approximation holds. Our
results demonstrate that questions regarding the validity of Migdal's
approximation go hand in hand with questions of the validity of a linear $e$-ph
interaction.
