  Let $F$ be a field of characteristic $p$. We show that
$\Hom_{F\Sigma_n}(S^\lambda, S^\mu)$ can have arbitrarily large dimension as
$n$ and $p$ grow, where $S^\lambda$ and $S^\mu$ are Specht modules for the
symmetric group $\Sigma_n$. Similar results hold for the Weyl modules of the
general linear group. Every previously computed example has been at most
one-dimensional, with the exception of Specht modules over a field of
characteristic two. The proof uses the work of Chuang and Tan, providing
detailed information about the radical series of Weyl modules in Rouquier
blocks.
