  The electronic structure of condensed matter can be significantly affected by
the electron-phonon interaction, leading to important phenomena such as
electrical resistance, superconductivity or the formation of polarons. This
interaction is often neglected in band structure calculations but can have a
strong impact on band gaps or optical spectra. Commonly used frameworks for
electron-phonon energy corrections are the Allen-Heine-Cardona theory and the
Fr\"ohlich model. While the latter shows qualitative agreement with experiment
for many polar materials, its simplicity should bring hard limits to its
applicability in real materials. Improvements can be made by introducing a
generalized version of the model, which considers anisotropic and degenerate
electronic bands, and multiple phonon branches. In this work, we search for
trends and outliers on over a thousand materials in existing databases of
phonon and electron band structures. We use our results to identify the limits
of applicability of the standard Fr\"olich model by comparing to the
generalized version, and by testing its basic hypothesis of a large radius for
the polaronic wavefunction and the corresponding atomic displacement cloud.
Among our extended set of materials, most exhibit large polaron behavior as
well as validity of the perturbative treatment. For the valence band, there is
also a significant fraction of the materials for which the perturbative
treatment cannot be applied and/or for which the size of the self-trapping
region is close to the atomic repetition distance. We find a large variety of
behaviors, and employ much more accurate, fully ab initio Allen-Heine-Cardona
calculations to understand extreme cases, where the Fr\"ohlich model should
fail and unusually large zero-point renormalization energies occur.
