Summary: E. M. Azevedo and D. Gottlieb [Econometrica 85, No. 1, 67–105 (2017; Zbl 1420.91228)] (AG) define a notion of equilibrium that always exists in the Rothschild and Stiglitz (RS) model [M. Rothschild and J. E. Stiglitz, “Equilibrium in competitive insurance markets: an essay on the economics of imperfect information”, Q. J. Econ. 90, No. 4, 629–649 (1976; doi:10.2307/1885326)] of competitive insurance markets, provided costs are bounded. However, equilibrium predictions are fragile: introducing an infinitesimal mass of high-cost individuals discretely increases all prices and reduces coverage for all individuals. We study sensitivity w.r.t. cost bounds by considering sequences of economies with increasing upper bounds of cost, and determining whether their equilibria converge. We present sufficient conditions under which AG equilibrium exists when cost is unbounded. For simple insurance markets, we derive a necessary and sufficient condition for existence: surplus from insurance increases faster than linearly with expected cost. This condition is empirically common. If the condition fails, a higher bound on cost results in market unraveling: all prices diverge and, in the limit, an AG equilibrium does not exist. We use these results to show that the equilibrium for an insurance market with an unbounded continuum of types is characterized by a simple differential equation. We also provide examples of non-existence for a (single-product) market for lemons with unbounded cost.