In ordinary solids, material disorder is known to increase the size of the process zone in which stress concentrates at the crack tip, causing a transition from localized to diffuse failure. Here, we report experiments on disordered 2D lattices, derived from frictional particle packings, in which the mean coordination number $\Z$ of the underlying network provides a similar control. Our experiments show that tuning the connectivity of the network provides access to a range of behaviors from brittle to ductile failure. We elucidate the cooperative origins of this transition using a frictional pebble game algorithm on the original, intact lattices. We find that the transition corresponds to the isostatic value $\Z = 3$ in the large-friction limit, with brittle failure occurring for structures vertically spanned by a rigid cluster, and ductile failure for floppy networks containing non-spanning rigid clusters. Furthermore, we find that individual failure events typically occur within the floppy regions separated by the rigid clusters.