We present two empirical studies on exclusives, it-clefts, and pseudoclefts (i.e., identity statements with a definite description) in which the at-issue and not-at-issue content-a factor that has not been properly controlled for in prior experimental work on cleft exhaustivity-were teased apart systematically. The results show that violations of exhaustivity in it-clefts, a not-at-issue inference, patterned differently from the necessary presupposition failures of the not-at-issue semantic inferences. These findings pose a new experimental challenge to semantic accounts of exhaustivity in it-clefts, while being in line with pragmatic accounts.