As a mathematical model for energy levels produced by impurities in a crystal, we study perturbations of a (periodic) Schrödinger operator $H =-\Delta + V$ by a potential $\lambda W$, where $\lambda$ is a real coupling constant and $W$ decays at infinity. Assuming that $H$ has a spectral gap, we ask for the number of eigenvalues which are moved into the gap and cross a fixed level $E$ in the gap, as $\lambda$ increases. Such “impurity levels” are a basic ingredient in the quantum mechanical theory of the color of crystals (insulators) and of the conductivity of (doped) semi-conductors in solid state physics. In the general case where $W$ is allowed to change its sign, we discuss upper and lower asymptotic bounds for the eigenvalue counting function. We also provide bounds for the total number of eigenvalues crossing $E$ as the height of a repulsive “barrier”, living on a compact set $K$, tends to $\infty$. While quasi- ical arguments give some useful hints, it turns out that, in particular, lower bounds are very sensitive and depend highly on the structure of the set $K$. Here decoupling via natural Dirichlet boundary conditions tends to play a dominating role, e.g. if the set $K$ has many small holes (“swiss cheese”).