Your discomfort is related to the theory of vagueness in philosophy. Statisticians generally believe that cases like yours are resolvable, and thus, this situation is a case of ambiguity rather than true vagueness (although this is ultimately a philosophical belief rather than something that can be proven). So, from a statistical perspective, we say that you simply have insufficient power, as standard logic (crisp sets) demands that A is either drawn from the same population as B, as C, or is drawn from it's own population. Thus, you must have at least 1 type II error. That is, if A is drawn from the same population as B, then $\text H:A=C$ should have been rejected, likewise if it's the same as C, then $\text H:A=B$ should have been rejected, and if drawn from a distinct population, then both nulls should have been rejected. In the interim, you can say that there is a difference between B and C. (Sorry to get all metaphysical.)