Recently, Ashby and Townsend (1986) introduced the General Recognition Theory as a theory of perceptual identification. Chief among their contributions were rigorous definitions of various types of perceptual independence of stimulus dimensions. Previously these distinctions had been obscured in the literature. Ashby and Townsend charactenzed many of the relationships between the different types of independence in theoretical terms. Several theorems provided conditions under which experimenters could obtain support (or falsification) for distinct types of independence. Theorem 4 in Ashby and Townsend stated several assumptions, including optimality that purportedly were sufficient for a strong form of perceptual independence. I show that this theorem is false by counterexample. The existence of this special case illustrates the difficulty of testing sensitive internal constructs like perceptual independence when only grosser measures of observable behavior are available. First a review the basic concepts of the GRT and independence is provided, with a discussion the above limitation and its implications for testing internal concepts like independence. Included are simulation results that illustrate prospects for testing for independence in lieu of the counterexample.