We establish a relation between long-time strong instability and the existence (in a certain generic sense) of unbounded orbits for dynamical systems on a Banach space. We then discuss some consequences of this relation for nonlinear Schrödinger equations. Namely, we prove long-time strong instability of plane wave solutions for the cubic nonlinearity and the existence of unbounded orbits for certain nonlinearities that are close (but not quite equal) to the cubic one.