Doped organic semiconductors are critical to emerging device applications, including thermoelectrics, bioelectronics, and neuromorphic computing devices. It is commonly assumed that low conductivities in these materials result primarily from charge trapping by the Coulomb potentials of the dopant counterions. Here, we present a combined experimental and theoretical study rebutting this belief. Using a newly developed doping technique based on ion exchange, we prepare highly doped films with several counterions of varying size and shape and characterize their carrier density, electrical conductivity, and paracrystalline disorder. In this uniquely large data set composed of several classes of high-mobility conjugated polymers, each doped with at least five different ions, we find electrical conductivity to be strongly correlated with paracrystalline disorder but poorly correlated with ionic size, suggesting that Coulomb traps do not limit transport. A general model for interacting electrons in highly doped polymers is proposed and carefully parametrized against atomistic calculations, enabling the calculation of electrical conductivity within the framework of transient localization theory. Theoretical calculations are in excellent agreement with experimental data, providing insights into the disorder-limited nature of charge transport and suggesting new strategies to further improve conductivities.