The assumption that certain natural language expressions trigger presuppositions leads to a famous problem, the so-called projection problem, and over the last few decades several comprehensive (yet very different) semantic/pragmatic theories have been proposed in an effort to solve it. In this talk I'll attempt to demonstrate a problem for a particular kind of approach to the analysis of presuppositions and its concomitant solutions to the projection problem. The kind of theory in question is founded on three intuitively natural assumptions, namely (a) that presuppositions are effectively constraints on interpretation (typically in a semantic sense), (b) that presuppositionality is lexically encoded, and (c) that whether a particular complex construction triggers a presupposition should be determinable purely on the basis of the constituents of the complex, combinatory rules, and syntax. I will argue that these three assumptions jointly lead to a severe problem for the semantics of propositional attitude verbs and that--with these assumptions in the background--there is no hope of correctly predicting presupposition projection across multiple attitude ascriptions. Note, that I will not be assuming any familiarity with the notion of presupposition nor the literature on the projection problem, so the talk should be accessible to a fairly wide audience.