Attractive colloidal dispersions, suspensions of fine particles which aggregate and frequently form a space spanning elastic gel are ubiquitous materials used as thermal insulators, catalytic electrodes in fuel cells, and in many consumer care products and food stuffs. The colloidal networks in these materials can exist in a mode of free settling when the network weight exceeds its compressive yield stress. An equivalent state occurs when the network is held fixed in place and used as a filter through which the suspending fluid is pumped. In either scenario, hydrodynamic instabilities leading to loss of network integrity occur. Recent experimental observations have shown that the loss of integrity is associated with the formation of eroded channels (so-called streamers) through which the fluid flows rapidly. Brownian dynamics simulations of sedimenting and hydrodynamically interacting colloids in gels are used to examine the initiation and propagation of this instability. A simple phenomenological model is developed that describes dynamically the radial growth of a streamer due to erosion of the network by rapid fluid back flow. The model exhibits a finite-time blow-up -- the onset of catastrophic failure in the gel -- due to activated breaking of the inter-colloid bonds and captures teh main features of teh instability and collapse dynamics. Model results provide insight into engineering strategies for avoiding settling instabilities in networks meant to have long shelf-lives.