Since the pioneering work by Daniel McFadden in the 1970s and 1980s (McFadden, 1973, 1981, 1982, 1984) discrete (multinomial) response models based on utility maximization have become an important tool of empirical researchers. A key feature of these models is the specification of utilities associated with the alternatives in terms of choice characteristics and individual preferences. Various generalizations of the basic models have been developed to allow for heterogeneity in taste parameters and heterogeneity in product characteristics that is unobserved to the econometrician. In this paper we investigate how rich a specification of the unobserved components is needed to rationalize arbitrary patterns of choice data (generated by utility-maximizing behavior) in settings with many individual decision makers and large choice sets. We find that in general the model must include at least one unobserved choice characteristic. If, as in common, one restricts the utility function to be monotone in the unobserved choice characteristic, then up to two unobserved choice characteristics may be needed to rationalize the choice data. We illustrate the results using scanner data about yoghurt purchases, employing a Bayesian estimation strategy that is particularly well suited to dealing with multiple unobserved product characteristics. We find that the inclusion of two unobserved choice characteristics leads to more reasonable estimates of elasticities.