Suppose $\Delta$ is a pure simplicial complex on $n$ vertices having codimension $c = n-\dim(\Delta)-1$. Terai and Yoshida proved that if the number of facets of $\Delta$ is at least $\binom{n}{c}-2c+1$, then $\Delta$ is Cohen-Macaulay. We improve this result by showing that these hypotheses imply the stronger condition that $\Delta$ is vertex decomposable. Further, this is a sharp bound. If time permits, we see that this in turn gives interesting properties for the Alexander dual ideal of the complex. This talk is based on joint work with Anton Dochtermann, Jay Schweig, Adam Van Tuyl, and Russ Woodroofe.