(Regarding crystals and their vibrations...) Usually, when calculating the vibrational frequencies, one first calculates the second order energy derivatives within the Born-Oppenheimer approximation, where the vibrational frequency is proportional to the curvature of the potential (curvature being the second derivative of the potential with respect to moving nuclei, adiabatically, along their normal mode coordinate). Quasi-harmonic and harmonic approximations stop at the 2nd order adiabatic derivative.
Calculations of anharmonic vibrations usually don't feature the calculation of adiabatic derivatives of the potential beyond fourth order, because the magnitude of the error in the Born-Oppenheimer approximation becomes comparable to the magnitude of higher order derivatives. These are still anharmonic methods, because they include derivatives beyond second order. That higher than fourth order derivatives are not considered accurate in the adiabatic approximation gives one a hazy sense of how reliable it is.
It is rare for the Born-Oppenheimer approximation to "break down" in the description of solids at low temperatures. At high temperatures, it is also rare to observe this, because one has much larger sources of error in their calculation. You are correct that smaller nuclei are expected to show more variation from adiabatic behaviour, but this effect could be washed out in vibrational calculations due to anharmonicity caused by a non-quadratic potential energy surface. Hydrogen vibrates violently, for instance, and does not display motion as simply harmonic as its heavier counterparts. This is not (primarily due to) a breakdown of the Born-Oppenheimer approximation however, but a result of $F=ma$ giving more acceleration to smaller masses, and thus pushing the light atoms over the purely harmonic region of motion.