Are structural break models true switching models or long memory processes? The answer to this question remains ambiguous. In recent years, many papers have dealt with this problem. Some studies have shown that, under specific conditions, switching models and long memory processes can easily be confused. In this paper, using several generating models (the mean-plus-noise model, the stochastic permanent break model, the Markov switching model, the threshold autoregressive (TAR) model, the sign model, and the structural change model) and several estimation techniques (the Geweke–Porter–Hudak (GPH) technique, detrended fluctuation analysis (DFA), the exact local Whittle (ELW) method, and wavelet methods) we show that, even if the answer is quite simple in some cases, it can be mitigated in other cases. Using French and American inflation rates, we found that the most appropriate process that takes into account the important features of these series is a model that simultaneously combines changes in regimes and long memory behavior. The main result of this study indicates that estimating a long memory parameter without taking into account the presence of breaks in the data sets may lead to misspecification and hence to overestimating the true parameter.