It was proved by Laufer that the Dolbeault cohomology groups are either zero or infinite dimensional for any open subset of a Stein manifold. In this paper, the author proves an analogue for CR manifolds. Namely, he shows that the cohomology groups of the Cauchy-Riemann complexes are either zero or infinite dimensional for a sufficiently small open subset of a generic CR manifold in \(\mathbb{C}^n.\) The author also provides an example of a CR manifold in \(\mathbb{C}^n\) for which the corresponding global result does not hold.