I did this myself!
(At least, if you interpret "incorrect" as "probably not true" rather than "demonstrably false".)
In the early days of homotopy type theory, we were starting to use multiple universes in ways that, I believe, had rarely been done before in type theory. Since Coq didn't support universe polymorphism yet, the only way to formalize some results was using type-in-type. In particular, I used it when formalizing some of the initial work on modalities that eventually became the RSS paper.
One of the results I proved, at that time, was that an (idempotent monadic) modality is left exact if and only if the universe of modal types is modal. This was a really striking application of, and justification for, the notion of left exactness. In particular, it implies that the modal types for a left exact modality form a model of type theory on their own, which is useful when constructing higher-topos models of type theory since any topos is a left exact localization of a presheaf topos.
Unfortunately, when we later went back to write up and formalize these results correctly using universe polymorphism, I realized that the proof was invalid. When universe levels are tracked, what I had proven was that the type of modal types in one universe $U$ was null with respect to all $U$-small modally-connected types; but since $U$ is not $U$-small, that doesn't imply $U$ is modal — unless we assume the modality is generated by a $U$-small family, i.e. that it is $U$-accessible. (In this form, the result is Theorem 3.11 in RSS.)
I don't have a counterexample to the statement in the non-accessible case, but I doubt it is true. In fact, for a non-accessible modality there isn't necessarily a canonical way to extend a modality on one universe to a modality on larger universes, so if we have a lex modality on $U$-small types, it's not even obvious what it means to say that $U$ is modal.
Ever since this experience, I've been unconvinced by any argument along the lines of "type-in-type is okay as long as you don't do anything stupid". I'm willing to grant that what I did was stupid; the point is that it's possible to do stupid things by accident.