In many practical cases, the identification of a system is done in closed loop with some controller. In this paper, we show that the internal stability of the resulting model, in closed loop with the same controller, is not always guaranteed if this controller is unstable and/or nonminimum phase, and that the classical closed-loop prediction-error identification methods present different properties regarding this stability issue. With some of these methods, closed-loop instability of the identified model is actually guaranteed. This is a serious drawback if this model is to be used for the design of a new controller. We give guidelines to avoid the emergence of this instability problem; these guidelines concern both the experiment design and the choice of the identification method.