Flow drag reduction induced by chemical additives, more commonly called drag-reducing agents (DRAs), has been studied for many years, but few studies can manifest the mechanism of this phenomenon. In this paper, a new mathematical model is proposed to predict the upper limit of drag reduction with polymer DRAs in a turbulent pipe flow. The model is based on the classic finitely extensible nonlinear elastic-Peterlin (FENE-P) theory, with the assumption that all vortex structures disappear in the turbulent flow, i.e., complete laminarization is achieved. With this model, the maximum drag reduction by a DRA at a given concentration can be predicted directly with several parameters, i.e., bulk velocity of the fluid, pipe size, and relaxation time of the DRA. Besides, this model indicates that both viscosity and elasticity contribute to the drag reduction: before a critical concentration, both viscosity and elasticity affect the drag reduction positively; after this critical concentration, elasticity still works as before but viscosity affects drag reduction negatively. This study also proposes a correlation format between drag reduction measured in a rheometer and that estimated in a pipeline. This provides a convenient way of pipeline drag reduction estimation with viscosity and modulus of the fluids that can be easily measured in a rheometer.