Consider the semiclassical  limit, as the Planck constant $\hbar\to 0$, of   bound  states of a  one-dimensional  quantum particle in multiple potential wells separated by  barriers. We show that, for each eigenvalue of the Schr\"odinger operator, the Bohr-Sommerfeld quantization condition is satisfied at least for one potential well. The proof of this result relies on a study of real wave functions in a neighborhood of  a potential  barrier. We show that, at least from one side, the barrier fixes the phase of wave functions in the same way as a potential    barrier of infinite width. On the other hand, it turns out that for each well there exists an eigenvalue in a small  neighborhood of every point satisfying  the Bohr-Sommerfeld condition.