Tilted twin boundaries (TBs), whose plane normals are rotated at an angle from the parent grain axis, naturally occur in columnar-grained, nano-twinned (NT) metals. Here, using a combination of atomistic simulations and analytical modeling, we reveal that NT metals with the ideal, non-tilted TBs exhibit continuously increasing strength with decreasing twin thickness, and hence, no inverse twin thickness size effect on strength. In contrast, NT metals with tilted TBs exhibit an inverse size effect, and the critical twin thickness, below which strength decreases, increases as the TB-tilt angle increases. The analysis also identifies a critical value of TB tilt, for which strength becomes independent of twin thickness and is the weakest. The transition arises from a change in dislocation activity prevailing mostly on planes inclined to TBs to planes parallel to the TBs. These findings reveal a profound influence of TB tilt angle that could redirect the analysis and engineering of nano-twin structures.