Do lexical items have internal structure that contributes to, or determines, the stable interpretation of their potential hosts? One argument in favour of the claim that lexical items are so structured is that certain putative verbs appear to be 'impossible', where the intended interpretation of them is apparently precluded by the character of their internal structure. The adequacy of such reasoning has recently been debated by Fodor and Lepore and Johnson, but to no apparent resolution. The present paper argues that such 'impossible word arguments' for internal lexical structure, although not apodictic, do constitute inferences to the best explanation for such structure. Alternative explanations for the 'impossible words' are considered and rejected.