  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 B^0_q-\bar B^0_q mixings (q=d,s) are described by a single operator (\bar b
q)_{V-A} (\bar b q)_{V-A}, the lower bounds on the mass differences Delta
M_{s,d} are simply given by their Standard Model values (Delta M_{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 beta our bound is
satisfied in spite of gluino and neutralino contributions.
