  For every $n \geq 1$, let $(\mathrm{FW}_n)$ denote the fixed-point property
for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube
complexes of dimension $n$). In this article, we construct groups satisfying
$(\mathrm{FW}_n)$ but not $(\mathrm{FW}_{n+1})$. Our examples are virtually
free abelian and virtually graph products of finite groups. As an application
of our constructions, we obtain, for every $n \geq 1$, a group satisfying
$(\mathrm{FW}_n)$ but acting properly and cocompactly on a median graph of
cubical dimension $n+1$. Several conjectures and open questions are included.
