Propositional inquisitive logic is the limit of its n-bounded approximations. In the predicate setting, however, this does not hold anymore, as discovered by Ciardelli and Grilletti [11], who also found complete axiomatizations of n-bounded inquisitive logics InqBQn, for every fixed n. We introduce cut-free labelled sequent calculi for these logics. We illustrate the intricacies of schematic validity in such systems by showing that the well-known Casari formula is atomically valid in (a weak sublogic of) predicate inquisitive logic InqBQ, fails to be schematically valid in it, and yet is schematically valid under the finite boundedness assumption. The derivations in our calculi, however, are guaranteed to be schematically valid whenever a single specific rule is not used. © The Author(s) 2026.