Strong ionic interactions in concentrated ionic liquids is shown to result in significant correlations and deviations from ideal solution behavior. We use rigorous concentrated solution theory coupled with molecular dynamics simulations to compute and explain the unusual dependence of transport properties on cation concentration in the Na+-Pyr(13)(+)-FSI- ionic liquid electrolyte. When accounting for intra- and inter-species correlation, beyond the commonly used uncorrelated Nernst-Einstein equation, an anomalously low and even negative transference number emerges for x(NaFSI) lower than 0.2. With increasing concentration the transference number monotonically increases, approaching unity, while the total conductivity decreases as the system transitions to a state resembling a single-ion solid-state electrolyte. The degree of spatial ionic association is explored further by employing unsupervised single-linkage clustering algorithm. We emphasize that strong ion-ion coupling in the electrolyte significantly impacts the trade-off between key electrolyte transport properties, and consequently governs the power density of the battery.