My physic teacher told me that experimental deviations from the predictions of Coulomb's law occur at small separations because, being inverse square, Coulomb's law work best for larger values of r. Why is this the case?
Almost every source of electric field in our day life is more than an point-like, having complex structure on the distribution of charges. The naive aplication of the Coulumb's law assume that the sourcer is point-like or spherically symmetric. This is a good approximation if the dimensions of the sourcer is neglegible. However, when you get close to the sourcer, the structure (the charge distribution) start to be relevant, and you need to replace the naive Coulomb's law by the Coulumb's law applied in each charge of the distribtuion.
It is important to note that the law is still valid. Is the application of the law that need to be corrected by new inputs, new distribution of charges. This charges can form a continuum too. And applying the Coulomb's law to each piece of the continuum (using calculus) gives the right predictions, with controlled accurancy.
There is a systematic way to approach to corrections of the pointe-like and spherically symmetric approximation, the multipole expansion. The idea is to expand the potential field in terms of $1/r$. The next correction to the $1/r^2$ electric field is the dipole