Given the ubiquity of layering in environmental stratifications, an interesting example being double-diffusive staircase structures in the Arctic Ocean, we present the results of a joint theoretical and laboratory experimental study investigating the impact of multiple layering on internal wave propagation. We first present results for a simplified model that demonstrates the non-trivial impact of multiple layering. Thereafter, utilizing a weakly viscous linear model that can handle arbitrary vertical stratifications, we perform a comparison of theory with experiments. We conclude by applying this model to a case study of a staircase stratification profile obtained from the Arctic Ocean, finding a rich landscape of transmission behaviour.