Because the minimal spinor representation of the 5D Lorentz group is twice as large as that of the 4D Lorentz group, one might think that a fine-tuning in the fermion sector is needed in order to construct a Lorentz invariant higher dimensional theory which includes fermions. In addition, 5D SUSY would then require at least 8 supercharges, which corresponds to ${\cal N}=2$ supersymmetry in 4D. We will demonstrate that 4D ${\cal N}=1$ supersymmetry plus gauge invariance (with properly chosen matter content) is enough to ensure the existence of the additional supersymmetries in the continuum limit. This phenomenon of enhanced supersymmetry generation is related to the behavior of these models at low energies in a purely 4D context, in which ${\cal N}$=1 SUSY is enhanced to ${\cal N}$=2 on the moduli space, without the fine-tuning of parameters. However, in the presence of additional hypermultiplets the required superpotential does have to be tuned in the 4D theories. The analogous effect that we obtain here is that maintaining 5D Lorentz invariance will require the tuning of a superpotential coupling in the 4D lattice models. We present the explicit construction of these models which will give in the continuum limit the 5D ${\cal N}=1$ 