The method of successive substitution for phase equilibrium calculations can only converge to a minimum of the mixture Gibbs free energy, and it is commonly assumed that, given proper initial estimates, convergence will always occur. Unfortunately, although the exemptions are rare, the assumption is incorrect. Divergence, regardless of the quality of the initial estimate, can occur in mixtures that exhibit strong negative deviations from ideal solution behavior, such as polymer solutions and gas hydrates. Unless special precautions are taken, algorithms that implement successive substitution as part of the solution procedure are likely to fail on such mixtures. In this paper we present an analysis of the thermodynamic conditions that may lead to divergence of successive substitution and present some numerical examples. In addition we briefly discuss some methods capable of solving the appropriate equilibrium equations under conditions where successive substitution fails.