Concentrated single phase solid solutions, including medium- and high-entropy alloys, represent a new class of materials that have recently attracted significant interest due to exceptional functional and structural properties. Their fascinating properties are mainly attributed to the sluggish atomic-level diffusion and transport, but its controlling mechanisms are largely unknown and there is certain skepticism about its very existence. By using microsecond-scale molecular dynamics, on-the-fly and conventional kinetic Monte Carlo, we reveal the governing role of percolation effects and composition dependence of the vacancy migration energy in diffusion. Surprisingly, an increase of concentration of faster species (Fe) in face-centered cubic Ni-Fe alloy may decrease the overall atomic diffusion. Consequently, the composition dependence of tracer diffusion coefficient has a minimum near the site percolation threshold, similar to 20 at.%Fe. We argue that this coupled percolation and composition-dependent barriers for vacancy jumps within different subsystems in medium- and high-entropy alloys leads, indeed, to the sluggish diffusion. A fast method for preselecting materials with potentially desired properties is suggested.