We present very deep spectrophotometry of 14 bright extragalactic H II regions belonging to spiral, irregular, and blue compact galaxies. The data for 13 objects were taken with the High Resolution Echelle Spectrometer on the Keck I telescope. We have measured C II recombination lines in 10 of the objects and O II recombination lines in eight of them. We have determined electron temperatures from line ratios of several ions, especially those of low ionization potential. We have found a rather tight linear empirical relation between T e([N II]) and T e([O III]). We have found that O II lines give always larger abundances than [O III] lines. Moreover, the difference of both O++ abundance determinations—the so-called abundance discrepancy factor—is very similar in all the objects, with a mean value of 0.26 ± 0.09 dex, independent of the properties of the H II region and of the parent galaxy. Using the observed recombination lines, we have determined the O, C, and C/O radial abundance gradients for three spiral galaxies: M33, M101, and NGC 2403, finding that C abundance gradients are always steeper than those of O, producing negative C/O gradients across the galactic disks. This result is similar to that found in the Milky Way and has important implications for chemical evolution models and the nucleosynthesis of C.