Suppose that E \mathcal {E} is a vector bundle on a smooth projective variety X X . Given a family of curves C C on X X , we study how the Harder-Narasimhan filtration of E | C \mathcal {E}|_{C} changes as we vary C C in our family. Heuristically we expect that the locus where the slopes in the Harder-Narasimhan filtration jump by μ \mu should have codimension which depends linearly on μ \mu . We identify the geometric properties which determine whether or not this expected behavior holds. We then apply our results to study rank 2 2 bundles on P 2 \mathbb {P}^{2} and to study singular loci of moduli spaces of curves.