Some extraordinary solutions of polymers in single solvents exhibit two liquid/liquid critical points, one within the usual composition range and a second one in the range of moderate to high polymer concentration. This particularity was studied theoretically by means of a recently established approach, which accounts explicitly for chain connectivity and for the ability of polymer molecules to respond to changes in their environment by conformational rearrangements. On the basis of model calculations, it is shown that the anomalous demixing behavior, i.e., the emergence of a second critical point, is bound to the necessary but not sufficient condition that a thermodynamically very unfavorable contact formation between polymer segments and solvent molecules (at fixed conformation of the components) is associated with an extraordinarily favorable conformational response. Possibilities to forecast whether certain systems behave normal or in an anomalous manner are discussed.