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Lexical ambiguity--the phenomenon of a single word having multiple, distinguishable senses--is pervasive in language. Both the degree of ambiguity of a word (roughly, its number of senses) and the relatedness of those senses have been found to have widespread effects on language acquisition and processing. Recently, distributional approaches to semantics, in which a word's meaning is determined by its contexts, have led to successful research quantifying the degree of ambiguity, but these measures have not distinguished between the ambiguity of words with multiple related senses versus multiple unrelated meanings. In this work, we present the first assessment of whether distributional meaning representations can capture the ambiguity structure of a word, including both the number and relatedness of senses. On a very large sample of English words, we find that some, but not all, distributional semantic representations that we test exhibit detectable differences between sets of monosemes (unambiguous words; N = 964), polysemes (with multiple related senses; N = 4,096), and homonyms (with multiple unrelated senses; N = 355). Our findings begin to answer open questions from earlier work regarding whether distributional semantic representations of words, which successfully capture various semantic relationships, also reflect fine-grained aspects of meaning structure that influence human behavior. Our findings emphasize the importance of measuring whether proposed lexical representations capture such distinctions: In addition to standard benchmarks that test the similarity structure of distributional semantic models, we need to also consider whether they have cognitively plausible ambiguity structure.