Using the change in conductance of a single nanometer-wide protein pore of the alpha-hemolysin channel to detect pore occupancy by polymers, we measure the equilibrium partitioning of differently sized poly(ethylene glycol)s as a function of polymer concentration in the bulk solution. In the semidilute regime, increased polymer concentration results in a sharp increase in polymer partitioning. Quantifying solution nonideality by osmotic pressure and taking the free energy of polymer confinement by the pore at infinite dilution as an adjustable parameter allows us to describe polymer partitioning only at low polymer concentrations. At larger concentrations the increase in partitioning is much sharper than the model predictions. The nature of this sharp transition between strong exclusion and strong partitioning might be rationalized within the concepts of scaling theory predicting this kind of behavior whenever the correlation length of the monomer density in the semidilute bulk solution becomes smaller than the pore radius. Specific attractive interactions between the protein pore and the polymer that exist in addition to the entropic repulsion accounted for in the present study may also play a role.