The question of whether features and behaviors that are characteristic to completely integrable systems persist in the transition to non-integrable settings is a central one in the study of nonlinear evolution equations. This issue is closely related to the broader problem of the stability of evolution equations. Another fundamental question concerns the lifespan of solutions: whether it is infinite or finite distinguishes between global-in-time existence and instability phenomena, the latter manifested as blow-up in finite time. We examine these questions in the context of the Nonlinear Schrödinger Equation (NLS) and NLS-type lattices, supplemented with nonzero boundary conditions at infinity. Numerical investigations, based on high-accuracy schemes, highlight the relevance of the accompanying mathematical analysis and yield numerical results in excellent agreement with theoretical predictions.