Given a family of rotationally symmetric compact manifolds indexed by the dimension and a weight function, the goal of this paper is to investigate the cut-off phenomenon for the Brownian motions on this family. We provide a class of compact manifolds with non-negative Ricci curvatures for which the cut-off in separation with windows occurs, in high dimension, with different explicit mixing times. We also produce counter-examples, still with non-negative Ricci curvatures, where there are no cut-off in separation. In fact we show a phase transition for the cut-off phenomenon concerning the Brownian motions on a rotationally symmetric compact manifolds. Our proof is based on a previous construction of a sharp strong stationary times by the authors, and some quantitative estimates on the two first moments of the covering time of the dual process. The concentration of measure phenomenon for the above family of manifolds appears to be relevant for the study of the corresponding cut-off.