The remnant of SN 1006 has an X-ray spectrum dominated by nonthermal emission, and pre-ASCA observations were well described by a synchrotron calculation with electron energies limited by escape. We describe the results of a much more stringent test: fitting spatially integrated ASCA GIS (0.6-8 keV) and RXTE PCA (3-10 keV) data with a combination of the synchrotron model SRESC newly ported to XSPEC and a new thermal shock model VPSHOCK. The new model can describe the continuum emission above 2 keV well, in both spatial distribution and spectrum. We find that the emission is dominantly nonthermal, with a small but noticeable thermal component: Mg and Si are clearly visible in the integrated spectrum. The synchrotron component rolls off smoothly from the extrapolated radio spectrum, with a characteristic rolloff frequency of 3.1×1017 Hz, at which the spectrum has dropped about a factor of 6 below a power-law extrapolation from the radio spectrum. Comparison of TeV observations with new TeV model images and spectra based on the X-ray model fits gives a mean postshock magnetic field strength of about 9 μG, implying (for a compression ratio of 4) an upstream magnetic field of 3 μG, and fixing the current energy content in relativistic electrons at about 7×1048 ergs, resulting in a current electron-acceleration efficiency of about 5%. This total energy is about 100 times the energy in the magnetic field. The X-ray fit also implies that electrons escape ahead of the shock above an energy of about 30 TeV. This escape could result from an absence of scattering magnetohydrodynamic waves above a wavelength of about 1017 cm. Our results indicate that joint thermal and nonthermal fitting, using sophisticated models, will be required for analysis of most supernova-remnant X-ray data in the future.