We define a notion of local overlaps in polyhedron unfoldings. We use this concept to construct new examples of polyhedra that cannot be edge unfolded without overlap. In particular, we present a polyhedron with 16 triangular faces for which every unfolding contains a vertex with total face angle greater than 2. We also construct an ununfoldable polyhedron with 9 convex faces, improving upon the previously best known bound of 13.