The plasmonic properties of bimetallic (gold and silver) core-shell nanostructures were studied numerically by using the boundary element method implemented in MATLAB (MNPBEM toolbox). The dependence of the bulk refractive index sensitivity of the nanostructures on the core/shell thicknesses was investigated, and regions of enhanced sensitivities were identified for both Ag@Au and Au@Ag type of nanostructures. For Ag@Au nanoparticles, utilizing a 14 ± 2 nm silver core radius with 4 ± 2 nm gold shell thickness or a 22 ± 2 nm core radius with 6 ± 1 nm shell thickness can yield a 1.2–2.5× increase in sensitivity compared to a purely silver nanosphere with the same size. For Au@Ag structures, 20 ± 3 nm gold core size with 10 ± 3 nm silver shell thickness was found to be optimal, with sensitivity enhancements between 2 and 5× compared to pure gold nanoparticles with the same size. The increased sensitivity can always be attributed to the hybridization of the plasmon absorbance peaks corresponding to the silver and gold parts, which merging comes with increased peak width (and thus decreased figure of merit) as a tradeoff. The provided sensitivity maps can be helpful for the design and fabrication of plasmonic sensors utilizing core-shell nanostructures.