  We analyze the decomposition rank (a notion of covering dimension for nuclear
$C^*$-algebras introduced by E. Kirchberg and the author) of subhomogeneous
$C^*$-algebras. In particular we show that a subhomogeneous $C^*$-algebra has
decomposition rank $n$ if and only if it is recursive subhomogeneous of
topological dimension $n$ and that $n$ is determined by the primitive ideal
space. As an application, we use recent results of Q. Lin and N. C. Phillips to
show the following: Let $A$ be the crossed product $C^*$-algebra coming from a
compact smooth manifold and a minimal diffeomorphism. Then the decomposition
rank of $A$ is dominated by the covering dimension of the underlying manifold.
