This paper investigated the nonlinear vibrations of an uncertain overhung rotor system with rub-impact fault. As the clearance of the rotor and stator is getting smaller, contact between them often occurs at high rotation speeds. Meanwhile, inherent uncertainties in the rubbing can be introduced for a variety of reasons, and they are typically restricted to small-sample variables. It is important to gain a robust understanding of the dynamics of such a system under non-probabilistic uncertainties. A non-intrusive uncertainty quantification scheme, coupled with the Runge-Kutta method, was used to study the effects of the rub-impact related interval uncertainties on the dynamical response individually and simultaneously, including the uncertainties in the contact stiffness, clearance, and friction coefficient. Moreover, the numerical validation of the developed analysis method was verified through comparisons with the scanning approach. The results obtained provide some guidance for investigating the uncertain dynamics of rubbing rotors and diagnosing the rub-impact fault under non-random uncertainty.