We demonstrate an improvement by more than 1 order of magnitude of the figure of merit (FoM) of plasmonic nanoparticle sensors by means of the diffractive coupling of localized surface plasmon resonances. The coupling in arrays of nanoparticles leads to Fano resonances with narrow line widths known as surface lattice resonances, which are very suitable for the sensitive detection of small changes in the refractive index of the surroundings. We focus on the sensitivity to the bulk refractive index and find that the sensor FoM scales solely with the frequency difference between the surface lattice resonance and the diffracted order grazing to the surface of the array. This result, which can be extended to other systems with coupled resonances, enables the design of plasmonic sensors with a high FoM over broad spectral ranges with unprecedented accuracy.