The interplay between shear and bulk viscosities on the flow harmonics, vn's, at the BNL Relativistic Heavy Ion Collider is investigated using the newly developed relativistic 2+1 hydrodynamical code v-usphydro that includes bulk and shear viscosity effects both in the hydrodynamic evolution and at freeze-out. While shear viscosity is known to attenuate the flow harmonics, we find that the inclusion of bulk viscosity decreases the shear viscosity-induced suppression of the flow harmonics, bringing them closer to their values in ideal hydrodynamical calculations. Depending on the value of the bulk viscosity to entropy density ratio, ζ/s, in the quark-gluon plasma, the bulk viscosity-driven suppression of shear viscosity effects on the flow harmonics may require a reevaluation of the previous estimates of the shear viscosity to entropy density ratio, η/s, of the quark-gluon plasma previously extracted by comparing hydrodynamic calculations to heavy-ion data.