Micropillar compression testing with repeated jumps in strain rate is used to circumvent inherent difficulties associated with nanoindentation and tensile testing of free-standing films. Application to sputtered 21 nm/21 nm Cu/Ni multilayer thin films with a cube-on-cube texture reveals an average strain rate sensitivity (m = 0.014) and activation volume (V = 17 b3), comparable to nanocrystalline face-centered cubic metals. Yet, m increases by ∼50% and V decreases by 70% with increasing strain, opposite to trends reported for nanotwinned Cu. The large, strain-dependent shifts in m and V are dependent on the underlying misfit dislocation structure of Cu/Ni interfaces.