  The symplectic Brill--Noether locus ${\mathcal S}_{2n, K}^k$ associated to a
curve $C$ parametrises stable rank $2n$ bundles over $C$ with at least $k$
sections and which carry a nondegenerate skewsymmetric bilinear form with
values in the canonical bundle. This is a symmetric determinantal variety whose
tangent spaces are defined by a symmetrised Petri map. We obtain upper bounds
on the dimensions of various components of ${\mathcal S}_{2n, K}^k$. We show
the nonemptiness of several ${\mathcal S}_{2n, K}^k$, and in most of these
cases also the existence of a component which is generically smooth and of the
expected dimension. As an application, for certain values of $n$ and $k$ we
exhibit components of excess dimension of the standard Brill--Noether locus
$B^k_{2n, 2n(g-1)}$ over any curve of genus $g \ge 122$. We obtain similar
results for moduli spaces of coherent systems.
