Room-temperature ionic liquids (RTILs) based on the N-butyl-N-methyl pyrrolidinium cation (PYR14+) combined with three different fluorinated anions have been prepared and characterized by NMR, conductivity, and rheological measurements. The anions are (trifluoromethanesulfonyl)(nonafluorobutanesulfonyl)imide (IM14-), bis(pentafluoroethanesulfonyl)imide (BETI-), and bis(trifluoromethanesulfonyl)imide (TFSI-). Intermolecular anion-cation nuclear Overhauser enhancements (NOEs) have been experimentally observed in all titled compounds. These findings indicate the formation of long-lived aggregates in the bulk liquids. The NOE patterns show marked selectivity and can be rationalized assuming that the perfluorinated moieties of the anions tend to adopt a preferential orientation with respect to the cations, with possible formation of mesoscopic fluorous domains. Self-diffusion coefficients D for the anion and the cation have been measured by DOSY NMR. Diffusion data show similar but not identical values for cation and anion, consistent with local ordering at the molecular level. The observed trend in diffusion coefficients, D-cation > D-anion for all compounds, is compatible with a higher degree of intermolecular organization of the anions. This nanoscale organization is connected to rather strong deviations of the experimental conductivities from those estimated from the ion diffusion coefficients through the Nernst-Einstein relationship. The measured viscosities and ion diffusion coefficients in PYR14IM14 and in PYR14TFSI have similar temperature dependencies, leading to very close values of the activation energies for these processes. Ab initio density functional calculations on models of a PYR14TFSI ion pair lead to the identification of several local minima, whose structure and energy can be qualitatively related to the experimental NOE signals and activation energies.