The present paper discusses novel data which are problematic for assertability conditions based on redundancy (Stalnaker 1979, Fox 2008, Schlenker 2009, Singh 2007, Chierchia 2009, Meyer 2013, Katzir & Singh 2014 among others). The problem comes from disjunctions like Either Mary isn’t pregnant or (she is and) it doesn’t show and in particular from the optional presence of she is (pregnant). These data are even more puzzling if compared to corresponding conditionals like If Mary is pregnant, (#she is and) it doesn’t show where the she is (pregnant) part is unacceptable as expected. In response to this puzzle, we present a solution based on two ingredients: (i) exhaustification and (ii) a notion of incremental redundancy. As we show, exhaustifying a sentence has an effect on the (incremental) redundancy status of its constituents. As a consequence of this, she is (pregnant) is actually not redundant in the disjunctive sentence above, provided the latter is exhaustified. We explore two possible ways of implementing this solution. The first is based on a definition of incremental redundancy which does not make use of local contexts as proposed by Fox (2008, 2013), building on Schlenker 2008. The second is based on Schlenker’s (2009) incremental theory of local contexts. We then briefly compare the two implementations and point to a potential advantage of the one based on local contexts in dealing with the different readings of the disjunctive sentence above.