Mass surveillance targets many users at the same time with the goal of learning as much as possible. Intuitively, breaking many users’ cryptography simultaneously should be at least as hard as that of only breaking a single one, but ideally security degradation is gradual: an adversary ought to work harder to break more. Bellare, Ristenpart and Tessaro (Crypto’12) introduced the notion of multi-instance security to capture the related concept for password hashing with salts. Auerbach, Giacon and Kiltz (Eurocrypt’20) motivated the study of public key encryption (PKE) in the multi-instance setting, yet their technical results are exclusively stated in terms of key encapsulation mechanisms (KEMs), leaving a considerable gap. We investigate the multi-instance security of public key encryption. Our contributions are twofold. Firstly, we define and compare possible security notions for multi-instance PKE, where we include PKE schemes whose correctness is not perfect. Secondly, we observe that, in general, a hybrid encryption scheme of a multi-instance secure KEM and an arbitrary data encapsulation mechanism (DEM) is unlikely to inherit the KEM’s multi-instance security. Yet, we show how with a suitable information-theoretic DEM, and a computationally secure key derivation function if need be, inheritance is possible. As far as we are aware, ours is the first inheritance result in the challenging multi-bit scenario.