The phase boundaries of polymer solutions in mixed solvents can be extremely complex due to the many competing van der Waals and associative interactions that can arise in these ubiquitous and technologically important complex fluids. The present paper focuses specific attention on ternary solutions of polymers (B) dissolved in a mixture of two solvents (A, C) that competitively associate with the polymer. We are particularly concerned with explaining the origin of the peculiar phenomenon of cononsolvency in mixed solvents, where a mixture of two individually good solvents behaves effectively as a poor solvent. Our computations are based on a recently developed generalization of Flory-Huggins theory that incorporates the competitive solvation of a polymer by two associating solvents. On the basis of this framework, we evaluate the limit of polymer phase stability (spinodal curves) and the second osmotic virial coefficient B-2 as a function of temperature and the composition of the pure solvent mixture that is maintained in osmotic equilibrium with the ternary solution. The calculations provide new insights into the miscibility patterns of ternary A/B/C polymer solutions exhibiting cononsolvency.