<p>Coated laminates arise from sequential lamination, where a core phase is repeatedly embedded into a coating phase along varying lamination directions and volume fractions. For non-degenerate material parameters, explicit formulas for the effective conductivity and stiffness of such microstructures are well established. This work investigates coated laminates with a <i>degenerate core</i>, including insulating and perfectly conducting phases in thermal conductivity as well as porous and rigid phases in linear elasticity. Motivated by observations in deep material networks, we analyze under which conditions the effective tensors of a coated laminate become non-degenerate despite the degeneracy of the core. We develop an abstract operator-theoretic framework in a Hilbert space setting that encompasses both conductivity and elasticity and allows a unified treatment of vanishing and infinite material parameters. Within this framework, we establish well-posedness of the coating process and derive explicit criteria for non-degeneracy in terms of intersection properties of suitable subspaces. The abstract results are subsequently specialized to thermal conductivity and linear elasticity, where the derived criteria yield explicit conditions for the non-degeneracy.</p>