Nominals assigned non-structural Case in English are prohibited from undergoing certain kinds of A-movement, in contrast to nominals assigned structural Case which are not so restrained. To account for the observed asymmetries, I propose that non-structural Case on a nominal undergoing A-movement must be suppressed and that non-structural Case suppression requires a locally c-commanding, lexically specified licensing head. I show that the inventory of heads licensing Case suppression can vary between idiolects and across non-structural Cases being suppressed, demonstrating the need for a lexically driven approach. I argue that non-structural Case suppression is accounted for with Remove (Müller 2017): certain heads can eliminate the non-structural K(ase)P shells projected above nominals, thereby allowing those nominals to A-move and to be assigned structural Case.