When can this condition on linear codes be satisfied?

Any maximal $r$-separated set is an $r$-net, otherwise you can augment it with a non-covered point.

But the same holds for linear codes! If a subspace $V$ is $r$-separated but not an $r$-net, you may take a far point $u$: then $\langle V,u\rangle$ is still $r$-separated. So a maximal $V$ fits.