We study a theory of session types in which we add a liveness property to the familiar safety one. In this setting, some subtype relations between session types that hold in other theories and that are commonly regarded as harmless become unsound. We present various equivalent definitions of the subtyping relation, we relate it with the standard ones, and we give algorithms for deciding it. Incidentally, we provide an original and remarkably simple coinductive characterization of the fair testing preorder for nondeterministic, sequential processes consisting of internal choices of outputs and external choices of inputs.