  From the general inverse theory of periodic Jacobi matrices, it is known that
a periodic Jacobi matrix of minimal period $p \geq 2$ may have at most $p-2$
closed spectral gaps. We discuss the maximal number of closed gaps for
one-dimensional periodic discrete Schr\"odinger operators of period $p$. We
prove nontrivial upper and lower bounds on this quantity for large $p$ and
compute it exactly for $p \leq 6$. Among our results, we show that a discrete
Schr\"odinger operator of period four or five may have at most a single closed
gap, and we characterize exactly which potentials may exhibit a closed gap. For
period six, we show that at most two gaps may close. In all cases in which the
maximal number of closed gaps is computed, it is seen to be strictly smaller
than $p-2$, the bound guaranteed by the inverse theory. We also discuss similar
results for purely off-diagonal Jacobi matrices.
