A subset $\mathcal{P}$ of $\mathbb{N}^2$ is called Schur
bounded if every infinite matrix with bounded scalar entries
which is zero off of $\mathcal{P}$ yields a bounded Schur
multiplier on $\mathcal{B}(\mathcal{H})$. Such sets are
characterized as being the union of a subset with at most $k$
entries in each row with another that has at most $k$ entries
in each column, for some finite $k$. If $k$ is optimal, there
is a Schur multiplier supported on the pattern with norm
$O(\sqrt k)$, which is sharp up to a constant. The same
characterization also holds for operator-valued Schur
multipliers in the cb-norm, i.e., every infinite matrix with
bounded \emph{operator} entries which is zero off of
$\mathcal{P}$ yields a completely bounded Schur
multiplier. This result can be deduced from a theorem of
Varopoulos on the projective tensor product of two copies of
$l^\infty$. Our techniques give a new, more elementary proof
of his result. We also consider the Schur multipliers for
certain matrices which have a large symmetry group. In these
examples, we are able to compute the Schur multiplier norm
exactly. This is carried out in detail for a few examples
including the Kneser graphs.