Predictions under common knowledge of payoffs may differ from those under arbi- trarily, but finitely, many orders of mutual knowledge; Rubinstein’s (1989) Email game is a seminal example. Weinstein and Yildiz (2007) showed that the discontinuity in the example generalizes: for all types with multiple rationalizable (ICR) actions, there exist similar types with unique rationalizable action. This paper studies how a wide class of departures from common belief in rationality impact Weinstein and Yildiz’s discontinuity. We weaken ICR to ICRλ, where λ is a sequence whose nth term is the probability players attach to (n − 1)th -order belief in rationality. We find that Weinstein and Yildiz’s discontinuity holds when higher-order belief in rationality remains above some threshold (constant λ), but fails when higher-order belief in rationality eventually becomes low enough (λ converging to 0).