We point out a class of trapping potentials in nonlinear Schrödinger equations that make them non-integrable, but prevent the emergence of power spectra associated with ergodicity. The potentials are characterized by equidistant energy spectra (e.g., the harmonic-oscillator trap), and therefore by a large number of resonances enhancing the nonlinearity. In a broad range of dynamical solutions, spanning the regimes of both weak and strong nonlinearity, the power spectra are shaped as narrow (quasi-discrete) evenly spaced spikes, unlike generic truly continuous (ergodic) spectra. We develop an analytical explanation for the emergence of these spectral features in the case of weak nonlinearity. In the strongly nonlinear regime, the presence of such structures is tracked numerically by performing simulations with random initial conditions. Some potentials that prevent ergodicity in this manner are of direct relevance to Bose-Einstein condensates: they naturally enter 1D, 2D and 3D Gross-Pitaevskii equations (GPEs), the quintic version of these equations, and a two-component GPE system.