We propose a framework for testing the possibility of large cascades in financial networks. This framework accommodates a variety of specifications for the probabilities of emergence of 'contagious links', where a contagious link leads to the default of a bank following the default of its counterparty. These are the first order contagion probabilities and depend on the shock propagation mechanism under consideration. When the cascade represents an insolvency cascade, and under complete observation of balance sheets, the first order contagion probabilities follow from the distribution of recovery rates. Under general contagion mechanisms and incomplete information, the financial network is modeled as an inhomogenous random graph in which only some of the banks' character-istics are observable. We give bounds on the size of the first order contagion and testable conditions for it to be small. For power-law financial networks, we also give a condition so that the higher order cascade dies out.