The achievable level of precision on photospheric abundances of stars is a major limiting factor on investigations of exoplanet host star characteristics, the chemical histories of star clusters, and the evolution of the Milky Way and other galaxies. While model-induced errors can be minimized through the differential analysis of spectrally similar stars, the maximum achievable precision of this technique has been debated. As a test, we derive differential abundances of 19 elements from high-quality asteroid-reflected solar spectra taken using a variety of instruments and conditions. We treat the solar spectra as being from unknown stars and use the resulting differential abundances, which are expected to be zero, as a diagnostic of the error in our measurements. Our results indicate that the relative resolution of the target and reference spectra is a major consideration, with use of different instruments to obtain the two spectra leading to errors up to 0.04 dex. Use of the same instrument at different epochs for the two spectra has a much smaller effect (~0.007 dex). The asteroid used to obtain the solar standard also has a negligible effect (~0.006 dex). Assuming that systematic errors from the stellar model atmospheres have been minimized, as in the case of solar twins, we confirm that differential chemical abundances can be obtained at sub-0.01 dex precision with due care in the observations, data reduction, and abundance analysis.