Nonlinear damping has been experimentally observed in nanoelectromechanical (NEMS) resonators and proposed as a path to enhanced quality (Q) factors. Furthermore, considering a nonlinearly damped Duffing NEMS resonator, it has been shown that white noise excitation (Brownian motion) can shrink the hysteresis region in the frequency-amplitude plot of the resonator response leading to higher Q factors. In this paper the authors: (1) derive an analytical expression using the method of harmonic balance for the frequency-amplitude relationship of a nonlinearly damped Duffing NEMS resonator and numerically validate the analytical result, (2) show that additive white Gaussian noise of increasing noise intensity, when superimposed on deterministic excitation can eliminate the hysteresis peaks implying higher Q factors and, (3) show that the aforementioned effects are much more pronounced in a resonator driven by Lévy flight excitation in contrast to Brownian excitation-the hysteresis peaks get eliminated at much lower levels of noise intensity with Lévy flight excitation. Lévy flights are non-Brownian stochastic processes characterized by occasional large departures from the mean. The results, in addition to providing insights into Q factor enhancement using Lévy flights, indicate the importance of non-Brownian noise processes in NEMS dynamics.