  We show that a Buchsbaum simplicial complex of small codimension must have
large depth. More generally, we achieve a similar result for ${\rm CM}_t$
simplicial complexes, a notion generalizing Buchsbaum-ness, and we prove more
precise results in the codimension 2 case. Along the paper, we show that the
${\rm CM}_t$ property is a topological invariant of a simplicial complex.
