At present, operators address the explosive growth of mobile data demand by densification of the cellular network so as to reduce the transmitter-receiver distance and to achieve higher spectral efficiency. Due to such network densification and the intense proliferation of wireless devices, modern wireless networks are interference-limited, which motivates the use of interference mitigation and coordination techniques. In this work, we develop a statistical framework to evaluate the performance of multi-tier heterogeneous networks with successive interference cancellation (SIC) capabilities, accounting for the computational complexity of the cancellation scheme and relevant network related parameters such as random location of the access points (APs) and mobile users, and the characteristics of the wireless propagation channel. We explicitly model the consecutive events of canceling interferers and we derive the success probability to cancel the n-th strongest signal and to decode the signal of interest after n cancellations. When users are connected to the AP which provides the maximum average received signal power, the analysis indicates that the performance gains of SIC diminish quickly with n and the benefits are modest for realistic values of the signal-to-interference ratio (SIR). We extend the statistical model to include several association policies where distinct gains of SIC are expected: (i) minimum load association, (ii) maxi- mum instantaneous SIR association, and (iii) range expansion. Numerical results show the effectiveness of SIC for the considered association policies. This work deepens the understanding of SIC by defining the achievable gains for different association policies in multi-tier heterogeneous networks.