The bulk and surface refractive index sensitivities of LSPR biosensors, consisting of coupled plasmonic nanosphere and nano-ellipsoid dimers, were investigated by simulations using the boundary element method (BEM). The enhancement factor, defined as the ratio of plasmon extinction peak shift of multi-particle and single-particle arrangements caused by changes in the refractive index of the environment, was used to quantify the effect of coupling on the increased sensitivity of the dimers. The bulk refractive index sensitivity (RIS) was obtained by changing the dielectric medium surrounding the nanoparticles, while the surface sensitivity was modeled by depositing dielectric layers on the nanoparticle in an increasing thickness. The results show that by optimizing the interparticle gaps for a given layer thickness, up to ~80% of the optical response range of the nanoparticles can be utilized by confining the plasmon field between the particles, which translates into an enhancement of ~3–4 times compared to uncoupled, single particles with the same shape and size. The results also show that in these cases, the surface sensitivity enhancement is significantly higher than the bulk RI sensitivity enhancement (e.g., 3.2 times vs. 1.8 times for nanospheres with a 70 nm diameter), and thus the sensors’ response for molecular interactions is higher than their RIS would indicate. These results underline the importance of plasmonic coupling in the optimization of nanoparticle arrangements for biosensor applications. The interparticle gap should be tailored with respect to the size of the used receptor/target molecules to maximize the molecular sensitivity, and the presented methodology can effectively aid the optimization of fabrication technologies.