Failures

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Assuming that electrons in completely filled bands do not contribute to the specific heat, how to explain the specific heat of insulators?
Actually the -term observed experimentally in the specific heat of many metals and non-metals cannot be attributed to the electrons.
The equilibrium size of solids which is connected only to the ions depends on temperature.
A melting solid is not static ...
The Wiedemann-Franz law law fails at intermediate temperatures.
What is the scattering mechanism that prevents normal metals from having infinite conductivity?
Electric insulators can conduct thermal current and sound.
Crystals show resonances in their reflectivity at energies far below the gap widths.
Scattering of laser light can show inelastic components.
The temperature-dependent intensity of X-ray diffraction and the background cannot be explained.

There are materials which show in some characteristic quantities a behaviour similar to the form predicted for the independent electron gas but of a size incompatible with the model, i. e. the density of states at the Fermi energy determined by measuring different quantities of the same material differs or cannot be brought into accordance with the measurements of band structures (Fermi liquids).
There are materials which show a behaviour of some characteristic quantities at low temperatures completely different from the IEG predictions.