The sensitivities of five different core-shell nanostructures were investigated towards changes in the refractive index of the surrounding medium. The shift of the localized surface plasmon resonance (LSPR) maximum served as a measure of the (respective) sensitivity. Thus, gold-silver core-shell nanoparticles (NPs) were prepared with different shell thicknesses in a two-step chemical process without the use of any (possibly disturbing) surfactants. The measurements were supported by ultramicroscopic images in order to size the resulting core-shell structures. When compared to sensitivities of nanostructures reported in the literature with those of the (roughly spherical) gold-silver core-shell NPs, the latter showed comparable (or even higher) sensitivities than gold nanorods. The experimental finding is supported by theoretical calculation of optical properties of such core-shell NP. Extinction spectra of ideal spherical and deformed core-shell NPs with various core/shell sizes were calculated, and the presence of an optimal silver shell thickness with increased sensitivity was confirmed. This effect is explained by the existence of two overlapping plasmon bands in the NP, which change their relative intensity upon change of refractive index. Results of this research show a possibility of improving LSPR sensor by adding an extra metallic layer of certain thickness.