Previous research has found that people can use word-object co-occurrences from ambiguous situations to learn word meanings (e.g., Yu & Smith, 2007). However, most studies of cross-situational learning present an equal number of words and objects, which may simplify the problem by, for example, encouraging learners to use assumptions such as each word going with one object. This paper presents several conditions in which the number of words and objects do not match: either additional objects appear at random, or objects appear sometimes without their intended words. These manipulations do generally hurt learning in comparison to balanced conditions, but people still learn a significant proportion of word-object pairings. The results are explored in terms of statistics of the training trials—including contextual diversity and context familiarity—and with the uncertainty- and familiarity-biased associative model. Parametric differences between conditions hint at hidden cognitive constructs.