Combining a variety of data with radiation hydrodynamic simulations, we examine the heating of the Sun's internetwork chromosphere and the hypothesis that the chromospheric '' basal '' emission arises because of acoustic wave dissipation. We focus on the 2s 2p 2 2 D 2s 2 2p 2 P o multiplet of C ii near 1335 Å , whose basal level of chromospheric emission has been reliably determined for stars and the Sun by Schrijver and colleagues. By accounting for center-to-limb variations and the different spectral bandpasses of the instruments used, we find that Schrijver's C ii solar basal intensity substantially exceeds stellar values, and that it can be identified with intensities seen in typical internetwork regions with the SUMER instrument on the SOHO spacecraft. Some time-series data sets of internetwork regions are then examined and compared with simulations made specifically for a typical observational data set, with vertical velocities at the lower boundaries fixed from observations with the MDI instrument on SOHO. The simulations can qualitatively account for the observed internetwork UV continuum fluctuations seen with SUMER, formed 0.6-0.85 Mm above the photosphere. However, they fail to capture almost any property of the observed internetwork C ii multiplet, which is formed substantially higher. The time-averaged simulations can account for between 1 7 and 1 4 of the C ii basal intensities; they predict oscillatory power between 5 and 10 mHz, whereas internetwork observations are dominated by low-frequency (<2 mHz) power of solar origin. The average simulated C ii intensities, which have a large contribution from the transition region heated by conduction down from a coronal upper boundary, fall short even of the smaller stellar basal intensities by a factor of !2. Together with known properties of weak, internetwork photospheric magnetic fields, we conclude that the internetwork upper chromosphere is probably dominated by magnetic heating. Thus, the solar basal (and internetwork) intensities of the C ii 1335 Å multiplet originate from magnetic, and not acoustic, mechanisms, in contradiction to the commonly accepted picture