We study the stability of network communication after removal of a fraction q=1-p of links under the assumption that communication is effective only if the shortest path between nodes i and j after removal is shorter than al(ij)(a> or =1) where l(ij) is the shortest path before removal. For a large class of networks, we find analytically and numerically a new percolation transition at p(c)=(kappa(0)-1)((1-a)/a), where kappa(0) [triple bond]<k(2)> / <k>and k is the node degree. Above p(c), order N nodes can communicate within the limited path length al(ij), while below p(c), N(delta) (delta<1) nodes can communicate. We expect our results to influence network design, routing algorithms, and immunization strategies, where short paths are most relevant.