The high axiality and Ising exchange interaction efficiently suppress quantum tunneling of magnetization of an asymmetric dinuclear Dy-III complex, as revealed by combined experimental and theoretical investigations. Two distinct regimes of blockage of magnetization, one originating from the blockage at individual Dy sites and the other due to the exchange interaction between the sites, are separated for the first time. The latter contribution is found to be crucial, allowing an increase of the relaxation time by 3 orders of magnitude.