Using Hubble Space Telescope ultraviolet and ground-based optical spectroscopy, we measure the C/O and N/O ratios of three metal-poor galaxies with similar metallicity but differing N/O. These observations, in conjunction with photoionization models, indicate that the C/O ratios estimated from C III] lambda 1909 and [O III] lambda 5007 lines are consistent with those measured from C II] lambda 2326 and [O II] lambda 3727 lines. Although inherently more uncertain than C++/O++ due to poorly known ionization correction factors, we develop the use of the C+/O+ ratio as a reasonable substitute diagnostic of the carbon abundance in H II regions. The C II] lambda 2326 multiplet is typically quite weak, but its proximity to the [O II] lambda lambda 3726, 3729 lines makes these transitions a potential tool for measuring the carbon abundance in high-redshift emission-line objects. The derived chemical properties are consistent with a statistically significant correlation between N and C abundances in metal-poor extragalactic H II regions, in the sense that systems with the largest N/O ratios also have the highest C/O ratios. This result is unexpected if the dispersion in N/O among galaxies of similar metallicity is caused by localized, temporary chemical enrichments from massive stars. The presence of a correlation suggests, instead, that the majority of N and C production is coupled, as expected from chemical evolution models where C is produced predominantly by low-mass stars and N is produced predominantly by intermediate-mass stars. The C/N ratio is, then, fixed by the initial mass function. Since the occurrence of localized chemical "pollution" in star-forming galaxies appears to be low the relative overabundance of N in some galaxies compared to others at similar metallicity is most plausibly interpreted as an indicator of the global, secular chemical enrichment history. As such, the N/O and perhaps the C/O ratios can be used as a "clock" to estimate the time since the last major episode of star formation.