Most novel noun-noun combinations are polysemous in that they tend to suggest several possible meanings. A finger cup can be a cup in which fingers are washed, a cup shaped like a finger, a narrow cup and so on. In this paper, we present a new theory of concept combination, the constraint theory, that accounts for the polysemy of noun-noun combinations. Constraint theory, which uses three constraints (of inclusion, plausibility and informativeness) acting over a unitary mechanism that generates candidate interpretations, makes certain predictions about the polysemy of different combinations. In particular, it predicts that combinations involving artifact terms should be more polysemous than those involving natural kinds because the former have functional models that promote multiple interpretations. In a single experiment, this prediction is confirmed along with other predictions about the types of interpretation that tend to be produced.