  We provide geometric methods to both lower- and upper-bound the large-scale
dimension of CAT(0) cube complexes quasiisometric to a given group $G$. When
these bounds overlap, this provides obstructions to $G$ being cocompactly
cubulated. More strongly, it prevents $G$ from being a coarse median space.
  As applications, we show that many free-by-cyclic groups cannot be
cocompactly cubulated, and that any tubular group with a coarse median is
virtually compact special.
