Let $F$ be a relatively closed subset of a Stein manifold. We prove that the $\overline{\partial }$-cohomology groups of Whitney forms on $F$ and of currents supported on $F$ are either zero or infinite dimensional. This yields obstructions of the existence of a generic $CR$ embedding of a CR manifold $M$ into any open subset of any Stein manifold, namely by the nonvanishing but finite dimensionality of some intermediate ${\overline{\partial }}_M$-cohomology groups.