Oscillator phase noise has been shown to be one of the main performance limiting factors in full-duplex systems. In this paper, we consider the problem of self-interference cancellation with phase noise suppression in full-duplex systems. The feasibility of performing phase noise suppression in full-duplex systems in terms of both complexity and achieved gain is analytically and experimentally investigated. First, the effect of phase noise on full-duplex systems and the possibility of performing phase noise suppression are studied. Two different phase noise suppression techniques with a detailed complexity analysis are then proposed. For each suppression technique, both free-running and phase-locked loop-based oscillators are considered. Due to the fact that full-duplex system performance highly depends on hardware impairments that are difficult to fully model, experimental results in a typical indoor environment are presented. The experimental results performed on two different platforms confirm results obtained from numerical simulations. Finally, the tradeoff between the required complexity and the gain achieved using phase noise suppression is discussed.