  We consider a wide class of closed convex cones $K$ in the space of real
$n\times n$ symmetric matrices and establish the existence of a chain of faces
of $K$, the length of which is maximized at $\frac{n(n+1)}{2} + 1$. Examples of
such cones include, but are not limited to, the completely positive and the
copositive cones. Using this chain, we prove that the distance to polyhedrality
of any closed convex cone $K$ that is sandwiched between the completely
positive cone and the doubly nonnegative cone of order $n \ge 2$, as well as
its dual, is at least $\frac{n(n+1)}{2} - 2$, which is also the worst-case
scenario.
