The semiclassical limit as the Planck constant h tends to 0 is considered for bound states of a one-dimensional quantum particle in multiple potential wells separated by barriers. It is shown that, for each eigenvalue of the Schrödinger operator, the Bohr-Sommerfeld quantization condition is satisfied for at least one potential well. The proof of this result relies on a study of real wave functions in a neighborhood of a potential barrier. It is shown that, at least from one side, the barrier fixes the phase of the 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.