Bilingualism is commonly assumed to improve creativity but the mechanisms underlying creative acts, and the way these mechanisms are affected by bilingualism, are not very well understood. We hypothesize that learning to master multiple languages drives individuals toward a relatively focused cognitive-control state that exerts strong top-down impact on information processing and creates strong local competition for selection between cognitive codes. Considering the control requirements posed by creativity tasks tapping into convergent and divergent thinking, this predicts that high-proficient bilinguals should outperform low-proficient bilinguals in convergent thinking, while low-proficient bilinguals might be better in divergent thinking. Comparing low- and high-proficient bilinguals on convergent-thinking and divergent-thinking tasks indeed showed a high-proficient bilingual advantage for convergent thinking but a low-proficient bilingual advantage for fluency in divergent thinking. These findings suggest that bilingualism should not be related to "creativity" as a unitary concept but, rather, to the specific processes and mechanisms that underlie creativity.