Two experiments investigated why preschool children sometimes produce multiple words for a referent (i.e. POLYNOMY), but other times seem to allow only one word. In Experiment 1, 40 three- and four-year-olds completed a modification of Deák & Maratsos' (1998) naming task. Although social demands to produce multiple words were reduced, children produced, on average, more than two words per object. Number of words produced was predicted by receptive vocabulary. Lexical insight (i.e. knowing that a word refers to function or appearance) and metalexical beliefs (i.e. that a hypothetical referent has one label, or more than one) were not preconditions of polynomy. Polynomy was independent of bias to map novel words to unfamiliar referents. In Experiment 2, 40 three- and four-year-olds learned new words for nameable objects. Children showed a correction effect, yet produced more than two words per object. Children do not have a generalized one-word-per-object bias, even during word learning. Other explanations (e.g. contextual restriction of lexical access) are discussed.