Hurford (1974) observed that a disjunction is generally infelicitous if one of the disjuncts entails the other, e.g., #John lives in Paris or he lives in France. Several accounts of Hurford’s observation have been put forward in the literature, grounding the infelicity of these so-called Hurford disjunctions into different principles of language use such as Logical Integrity (Anvari 2018, 2019), Mismatching Implicatures (Singh 2010, Meyer 2014), Non-Redundancy (Katzir & Singh 2013) or Non-Triviality (Schlenker 2009). In this paper, we investigate three variants of Hurford’s original case: (i) Hurford Disjunctions embedded in a downward entailing environment, e.g., #Everyone who lives in Paris or in France likes bread, (ii) felicitous disjunctions which we call Quasi-Hurford Disjunctions e.g., John lives in Paris or somewhere else in France and their clausal variants, e.g., John lives in France, or he lives in France but not in Paris, and (iii) disjunctions with extra constituents intervening between the entailing and the entailed disjuncts, e.g., #John lives in France, or (else) he lives in London or in Paris, which we dub Long-Distance Hurford Disjunctions. We show that none of the four accounts above captures all at once Hurford’s original case and the three variants above. As we discuss, the molecular approach by Chierchia (2009) and Katzir & Singh (2013) and the exhaustification-based approach by Mayr & Romoli (2016) can be combined with either the Non-Redundancy or the Non-Triviality to help with the overgeneration issues with Quasi-Hurford Disjunctions. As we show, however, the resulting theories can no longer account for the infelicity of Long-Distance Hurford Disjunctions. We conclude by summarizing the challenges raised by our data for existing approaches to informational oddness and, more broadly, for the descriptive generalization originally proposed by Hurford.