Monolingual children sometimes resist learning second labels for familiar objects. One explanation is that they are guided by word learning constraints that lead to the assumption that objects have only one name. It is less clear whether bilingual children observe this constraint. In the current study, we test the hypothesis that bilingual children might be more willing to accept second labels for objects and ask how they are affected by different amounts of information relevant to the second label. Although monolingual and bilingual children benefited from increased levels of information, only bilingual children chose the referent at above chance levels when they were offered increased levels of information. They were also more likely than monolingual children to accept second labels. Differences emerged even when English language vocabulary size was controlled for in the analyses.