  Given a group $G$ and an integer $n\geq 0$ we consider the family
$\mathcal{F}_n$ of all virtually abelian subgroups of $G$ of rank at most $n$.
In this article we prove that for each $n\ge2$ the Bredon cohomology, with
respect to the family $\mathcal{F}_n$, of a free abelian group with rank $k >
n$ is nontrivial in dimension $k+n$; this answers a question of Corob Cook,
Moreno, Nucinkis and Pasini. As an application, we compute the minimal
dimension of a classifying space for the family $\mathcal{F}_n$ for braid
groups, right-angled Artin groups, and graphs of groups whose vertex groups are
infinite finitely generated virtually abelian groups, for all $n\ge2$. The main
tools that we use are the Mayer-Vietoris sequence for Bredon cohomology,
Bass-Serre theory, and the L\"uck-Weiermann construction.
