We demonstrate that in the class of models with Constrained Minimal Flavour Violation (CMFV), in which the CKM matrix is the only source of CP violation and B⁰_q−B̅⁰_q mixings (q = d,s) are described by a single operator (b̅q)_V−A(b̅q)_V−A, the lower bounds on the mass differences ΔM_s,d are simply given by their Standard Model values (ΔM_s,d)_SM with two possible exceptions that we identify. Our proof involves all possible charged gauge boson, Goldstone boson and physical charged and neutral scalar exchanges and assumes that the masses of new charge +2/3 singlet heavy fermions T_i that mix with the top quark have all to be larger than m_t, which is chosen by nature. Similarly, the additional charged gauge bosons have to satisfy M_i>M_W, while no such bound needs to be set on the charged and neutral scalar masses. The two possible exceptions arise in the presence of U(1) neutral gauge bosons and Majorana fermions in box diagrams with flavour violating couplings. However in the case of the MSSM with CMFV and low tan β our bound is satisfied in spite of gluino and neutralino contributions.