Summary: In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of J. F. A. K. van Benthem [Theoria 45, 63–77 (1979; Zbl 0452.03013)], and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, \(\mathcal{V}\)-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to \(\mathcal{V}\)-baos, namely the provability logic \(GLB\) [G. K. Dzhaparidze, in: Proceedings of the 4th Soviet-Finnish symposium on logic, 1985. Tbilisi: Metsniereba. 16–48 (1988; Zbl 0725.03038); G. Boolos, The logic of provability. Cambridge: Cambridge University Press (1993; Zbl 0891.03004)]. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is \(\mathcal{V}\)-complete. After these results, we generalize the Blok Dichotomy [W. J. Blok, On the degree of incompleteness of modal logic and the covering relation in the lattice of modal logics. Techn. Rep., University of Amsterdam (1978)] to degrees of \(\mathcal{V}\)-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.