We propose a simple model that aims at describing, in a stylized manner, how local breakdowns due to imbalances or congestion propagate in real dynamical networks. The model converges to a self-organized critical stationary state in which the network shapes itself as a consequence of avalanches of rewiring processes. Depending on the model's specification, we obtain either single-scale or scale-free networks. We characterize in detail the relation between the statistical properties of the network and the nature of the critical state, by computing the critical exponents. The model also displays a nontrivial, sudden collapse to a complete network.