  We present novel previously unexplored periodic solutions, expressed in terms
of Jacobi elliptic functions, for both a coupled $\phi^4$ model and a coupled
nonlinear Schr\"odinger equation (NLS) model. Remarkably, these solutions can
be elegantly reformulated as a linear combination of periodic kinks and
antikinks, or as a combination of two periodic kinks or two periodic pulse
solutions. However, we also find that for $m=0$ and a specific value of the
periodicity (or at a nonzero value of the elliptic modulus $m$) this
superposition does not hold. These results demonstrate that the notion of
superposed solutions extends to the coupled nonlinear equations as well.
