This paper experimentally and theoretically investigates the fundamental bounds on radio localization precision of far-field Received Signal Strength (RSS) measurements. RSS measurements are proportional to power-flow measurements time-averaged over periods long compared to the coherence time of the radiation. Our experiments are performed in a novel localization setup using 2.4GHz quasi-monochromatic radiation, which corresponds to a mean wavelength of 12.5cm. We experimentally and theoretically show that RSS measurements are cross-correlated over a minimum distance that approaches the diffraction limit, which equals half the mean wavelength of the radiation. Our experiments show that measuring RSS beyond a sampling density of one sample per half the mean wavelength does not increase localization precision, as the Root-Mean-Squared-Error (RMSE) converges asymptotically to roughly half the mean wavelength. This adds to the evidence that the diffraction limit determines (1) the lower bound on localization precision and (2) the sampling density that provides optimal localization precision. We experimentally validate the theoretical relations between Fisher information, Cramér-Rao Lower Bound (CRLB) and uncertainty, where uncertainty is lower bounded by diffraction as derived from coherence and speckle theory. When we reconcile Fisher Information with diffraction, the CRLB matches the experimental results with an accuracy of 97-98%.