Epistemological disjunctivists such as Duncan Pritchard claim that in paradigmatic cases of knowledge the rational support for the known propositions is both factive and reflectively accessible. This position faces some problems, including the basis problem – how can our knowledge be based on such strong reasons that seem to leave no room for non-knowledge and therefore presuppose knowledge? – and the access problem – can disjunctivists avoid the implausible claim that we can achieve knowledge through inference from our introspective awareness of those reasons? I argue that disjunctivists cannot solve both of these problems at the same time by posing the dilemma question whether we can have factive and reflectively accessible reasons without knowledge. While I focus on Pritchard throughout most of the paper, I argue in the last section that other anti-skeptical versions of disjunctivism face the same dilemma.