Atomic data are an important source of systematic uncertainty in our determinations of nebular chemical abundances. However, we do not have good estimates of these uncertainties since it is very difficult to assess the accuracy of the atomic data involved in the calculations. We explore here the size of these uncertainties by using 52 different sets of transition probabilities and collision strengths, and all their possible combinations, to calculate the physical conditions and the total abundances of O, N, S, Ne, Cl, and Ar for a sample of planetary nebulae and H II regions. We find that atomic data variations introduce differences in the derived abundance ratios as low as 0.1$-$0.2 dex at low density, but that reach or surpass 0.6$-$0.8 dex at densities above 10$^{4}$ cm$^{-3}$ in several abundance ratios, like O/H and N/O. Removing from the 52 datasets the four datasets that introduce the largest differences, the total uncertainties are reduced, but high density objects still reach uncertainty factors of four for their values of O/H and N/O. We identify the atomic data that introduce most of the uncertainty, which involves the ions used to determine density, namely, the transition probabilities of the S$^{+}$, O$^{+}$, Cl$^{++}$, and Ar$^{+3}$ density diagnostic lines, and the collision strengths of Ar$^{+3}$. Improved calculations of these data will be needed in order to derive more reliable values of chemical abundances in high density nebulae. In the meantime, our results can be used to estimate the uncertainties introduced by atomic data in nebular abundance determinations.
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