We investigate the effects of a large (but finite) state space on models of efficient risk sharing. A group of risk-averse agents agree on a risk-sharing agreement in an economy without aggregate risk. The economy is subject to a perturbation, or shock, that prompts a renegotiation of the agreement. If agents insist on an $\ep$-utility improvement to accept a new agreement, then the probability of a post-shock acceptable agreement vanishes exponentially to zero as the number of states grows. We use similar arguments to consider a model where agents have multiple prior preferences, and show that the existence of an $\ep$-Pareto improving trade requires that some sets of priors have vanishingly small measure. Our results hinge on the "shape does not matter" message of high dimensional isoperimetric inequalities.