Drag reduction by polymers in turbulent wall-bounded flows exhibits universal and non-universal aspects. The universal maximal moan-velocity profile was explained in a recent theory. The saturation of this profile and the crossover back to the Newtonian plug are non-universal, depending on Reynolds number Re, concentration of polymer cp and the degree of polymerization N p. We explain the mechanism of saturation stemming from the finiteness of extensibility of the polymers, predict its dependence on c p and N in the limit of small cp and large Re, and present the excellent comparison of our predictions to experiments on drag reduction by DNA.