  We show that C*-algebras of the form C(X) \otimes Z, where X is compact and
Hausdorff and Z denotes the Jiang--Su algebra, have decomposition rank at most
2. This amounts to a dimension reduction result for C*-bundles with
sufficiently regular fibres. It establishes an important case of a conjecture
on the fine structure of nuclear C*-algebras of Toms and the second named
author, even in a nonsimple setting, and gives evidence that the topological
dimension of noncommutative spaces is governed by fibres rather than base
spaces.
