Materials with negative real effective permittivity and permeability (also known as "double-negative (DNG)" media) may exhibit interesting features that lead to unconventional phenomena in guidance of electromagnetic waves. As one such class of problems, in some of our earlier works we studied the guided wave propagation in structures containing a layer of DNG material paired with a slab of conventional (i.e., "double-positive (DPS)") material. In particular, we showed that such a DNG-DPS bilayer has unusual properties as a guided-wave structure. For instance, a parallel-plate waveguide filled with a pair of parallel lossless DNG and DPS layers may support dominant TE and TM modes that can propagate even when the waveguide total thickness is electrically very small. In the present work, we underline the possibility to explain some of the unconventional EM characteristics of such paired DPS-DNG waveguides and cavities using the distributed-circuit-element approach with appropriate choice of elements. The negative nature of the real part of permittivity and/or permeability affects the choice of such distributed circuit elements, and in turn the overall circuit structure exhibits certain natural "resonances" that may not be present when conventional materials are employed in such waveguides or cavities. The "circuit-element" approach can also be applied to "single-negative (SNG)" layers, in which only one of the material parameters, not both, has negative real part. Here, we first present a brief overview of our work on modal analysis in waveguides and cavities with DNG or SNG layers, and we then present our results on their distributed-circuit-element modeling.