Logics without or with weakened Universal Generalization?

Question

Are there any well-known/useful logics in which UG fails to hold in full generality? I'm developing a logic that well captures a phenomenon at the level of propositional logic (with no quantifier), but given the axioms, enriching the system with quantifiers cannot retain UG in its full generality. In my opinion, the nice propositional-fragment behaviour should make up for the failure of UG in quantificational fragments, but I was wondering if there have been similar situations to learn from.