  We prove that for m > 2, the m-th symbolic power of a Stanley-Reisner ideal
is Cohen-Macaulay if and only if the simplicial complex is a matroid.
Similarly, the m-th ordinary power is Cohen-Macaulay for some m > 2 if and only
if the complex is a complete intersection. These results solve several open
questions on the Cohen-Macaulayness of ordinary and symbolic powers of
Stanley-Reisner ideals. Moreover, they have interesting consequences on the
Cohen-Macaulayness of symbolic powers of facet ideals and cover ideals.
