  Multipliers of reproducing kernel Hilbert spaces can be characterized in
terms of positivity of $n \times n$ matrices analogous to the classical Pick
matrix. We study for which reproducing kernel Hilbert spaces it suffices to
consider matrices of bounded size $n$. We connect this problem to the notion of
subhomogeneity of non-selfadjoint operator algebras. Our main results show that
multiplier algebras of many Hilbert spaces of analytic functions, such as the
Dirichlet space and the Drury-Arveson space, are not subhomogeneous, and hence
one has to test Pick matrices of arbitrarily large matrix size $n$. To treat
the Drury-Arveson space, we show that multiplier algebras of certain weighted
Dirichlet spaces on the disc embed completely isometrically into the multiplier
algebra of the Drury-Arveson space.
