Lifetimes of excited states in the neutron-deficient odd-odd nucleus 166Re have been measured for the first time using the recoil distance Doppler-shift method. The measured lifetime for the (8−) state; τ = 480 (80) ps, enabled an assessment of the multipolarities of the γ rays depopulating this state. Information on electromagnetic transition strengths were deduced for the γ -ray transitions from the (9−), (10−), and (11−) states, and in the case of the (10−) and (11−) states limits on the B(M1) and B(E2) strengths were estimated. The results are compared with total Routhian surface predictions and semiclassical calculations. Tilted-axis cranking calculations based on a relativistic mean-field approach (TAC-RMF) have also been performed in order to test the possibility of magnetic rotation in the 166Re nucleus. While the TAC-RMF calculations predict a quadrupole-deformed nuclear shape with similar β2 deformation as obtained by using the TRS model, it was found that the experimental electromagnetic transition rates are in better agreement with a collective-rotational description.