  The conullity of a curvature tensor is the codimension of its kernel. We
consider the cases of conullity two in any dimension and conullity three in
dimension four. We show that these conditions are compatible with non-negative
sectional curvature only if either the manifold is diffeomorphic to
$\mathbb{R}^n$ or the universal cover is an isometric product with a Euclidean
factor. Moreover, we show that finite volume manifolds with conullity 3 are
locally products.
