The light absorption in light-harvesting complexes is performed by molecules such as chlorophyll, carotenoid, or bilin. Recent experimental findings in some of these complexes suggest the existence of long-lived coherences between the individual pigments at low temperatures. In this context, the question arises if the bath-induced fluctuations at different chromophores are spatially correlated or not. Here we investigate this question for the Fenna-Matthews-Olson (FMO) complex of Chlorobaculum tepidum by a combination of atomistic theories, i.e., classical molecular dynamics simulations and semiempirical quantum chemistry calculations. In these investigations at ambient temperatures, only weak correlations between the movements of the chromophores can be detected at the atomic level and none at the more coarse-grained level of site energies. The often-employed uncorrelated bath approximations indeed seem to be valid. Nevertheless, correlations between fluctuations in the electronic couplings between the pigments can be found. Depending on the level of theory employed, also correlations between the fluctuations of site energies and the fluctuations in electronic couplings are discernible.