The relation between the accuracy of centroid molecular dynamics correlation functions, and the geometry of the centroid potential is investigated. It is shown that, depending on the temperature, there exist several regimes, and in each of them certain features of the exact Kubo correlation functions are reproduced. The change of regimes is related to the emergence of barriers in the centroid potential. In order to clarify how the above described picture of regimes is modified in real systems when dissipation is important, a methodology is developed to test the accuracy of centroid correlation functions for the model of a particle coupled to a harmonic heat bath. A modification of the centroid molecular dynamics method to include the influence of the heat bath is introduced. Preliminary results of comparison of centroid molecular dynamics with the numerically exact results of filtered propagator functional method are presented.