Linewidths of the ESR spectra of the naphthalene, anthracene, and tetracene radical anions, and the tetracene cation in dilute solution at low temperatures have been carefully investigated. It was expected that for these systems the linewidth variations would be determined primarily by the anisotropic g tensor and electron-nuclear magnetic dipole interactions. A comparison of the experimental linewidth parameters with calculated values based on this assumption shows that although theoretical expressions can account qualitatively for the observed linewidths, quantitative agreement is not satisfactory. It is shown that the discrepancies between the calculated values and the experimental results cannot be accounted for by errors in the calculations resulting from our ignorance of the exact spin density distribution in the radicals or of the detailed form of the atomic-orbital wavefunction used for the unpaired electron. The relative insensitivity of the calculated ratios of linewidth parameters to changes in spin densities indicates that even if agreement between theory and experiment were excellent, it would not be possible to perform an indirect analysis of the experimental results to obtain reliable values for the spin densities. Both anisotropic rotational motions and fluctuations in the isotropic hyperfine splittings are found to affect the calculated linewidth parameters, and while neither of these factors alone is sufficient to explain the observed values, the calculations indicate that both of them may make significant contributions to the linewidths.