We give a group theoretic characterization of geodesics with superlinear
divergence in the Cayley graph of a right-angled Artin group A(G) with
connected defining graph G. We use this to determine when two points in an
asymptotic cone of A(G) are separated by a cut-point. As an application, we
show that if G does not decompose as the join of two subgraphs, then A(G) has
an infinite-dimensional space of non-trivial quasimorphisms. By the work of
Burger and Monod, this leads to a superrigidity theorem for homomorphisms from
lattices into right-angled Artin groups.