We report the results of near-infrared (1-2.5 μm) spectroscopy of three classical novae, V1819 Cyg (Nova Cygni 1986), V827 Her (Nova Herculis 1987), and V2214 Oph (Nova Ophiuchi 1988), observed from December 1986 to August 1989. Particular emphasis is placed on discussing the origin of the "coronal" emission. Extinction and distances are estimated for V1819 Cyg and V827 Her. The ratios of both the hydrogen recombination line, Brγ 2.167 μm, and the coronal line, [Si VI] 1.960 μm, to the continuum flux, are consistent with the hypothesis that the principal ejecta of the nova is cool (T∼104 K), favoring the idea that the observed coronal species in novae are primarily due to photoionization. If it is assumed instead that the coronal emission is due to collisional ionization in shock-heated gas, then our observations can be used to constrain the nature of the shock heating. Estimates of the coronal temperature based on the [Si VII]/[Si VI] line-intensity ratios indicate coronal temperatures of approximately log T = 5.73 for V827 Her and log T = 5.62 for V1819 Cyg. For both novae, the silicon ionization temperature remained nearly constant over a year. By using an emission measure derived from the [Si VI] line, we estimate the volume of hot gas necessary to produce the observed coronal emission in V1819 Cyg. We find that for coronal gas densities lower than 105 cm-3 the volume of shock-heated gas would exceed the volume that could have been swept out by the expanding shell of ejecta. We thus conclude that if the coronal region is truly hot gas, it must have a density greater than 105 cm-3. For coronal gas with a density this high, the cooling time is on the order of days. In order to maintain a constant coronal temperature over the period of a year, some steady source of heating is necessary. These observations should serve to constrain any models attempting to explain the observed coronal emission in terms of shock-heated gas. We discuss the prevalence of a coronal phase in the evolution of classical novae, and suggest some avenues to pursue in order to resolve the question of the ionization mechanism.