ESR measurements have been performed on samples of neutral polypyrrole, polypyrrole-perchlorate, oxygen-doped polypyrrole, and poly($\ensuremath{\beta}$,${\ensuremath{\beta}}^{\ensuremath{'}}$-dimethylpyrrole perchlorate). A narrow Lorentzian line is observed in oxidized, highly conducting films, but no correlation between conductivity and either linewidth or intensity is found. Electrochemically cycled films, which remain highly conducting, show little or no detectable ESR absorption. It is deduced that the ESR signal seen in the as-prepared material does not arise from the current-carrying species, but rather from accidental neutral $\ensuremath{\pi}$-radical defects. The absence of paramagnetism in the metallic state is discussed within the framework of bipolaron formation.