We show unexpected drag reduction as solvent flows past polymer-coated surfaces in solution. Adsorbed polymer layers of two types, poly(vinyl alcohol) (PVA) homopolymers in deionized water and poly(vinylpyridine)-polybutadiene diblock copolymers (PVP-PB) in tetradecane, were placed at variable spacings larger than twice the layer thickness within a modified surface forces apparatus, and the hydrodynamic forces owing to flow of solvent past these layers were measured as a function of surface spacing, pumping frequency, and pumping velocity (product of frequency and amplitude). When the flow rate was below a critical level, which depended on the system, the findings agreed with a simple hydrodynamic picture in which the solvent appeared to flow past surfaces of defined spacing, the solid-solid spacing less twice the "hydrodyamic radius" (R-H). The value of R-H was not always constant, however. In the PVA system it tripled as frequency was raised from 1 to 50 Hz though it was frequency-independent in the PVP-PB system. There was also strong dependence on flow rate. When the flow rate exceeded a critical level, the magnitude of hydrodynamic forces became up to an order of magnitude less than can be described by flow past a layer of thickness R-H and the "stick" boundary condition at that layer. A striking observation, which we present without definitive explanation at this time, is that when the flow rates were sufficiently high, the hydrodynamic forces decreased below those for flow of these solvents with no polymer layer at the surface and the "stick" boundary condition, The onset of this drag reduction occurred at the peak shear stress of 8 pN molecule(-1) (PVP-PB) or 12 pN molecule(-1) (PVA).