We present results of our Chandra/ACIS observations of the field centered on the fast, runaway O star AE Aur and its bow shock. Previous XMM-Newton observations revealed an X-ray "blob" near the IR arc tracing the bow shock, possibly a nonthermal source consistent with models of Inverse Compton scattering of dust IR photons by electrons accelerated at the shock. The new, subarcsecond-resolution Chandra data, while confirming the presence of the XMM-Newton source, clearly indicate that the latter is neither extended nor coincident with the IR arc and strongly suggest it is a background active galactic nucleus. Motivated by results published for the bow shock of BD+43{deg}3654, we extended our study to the radio domain by analyzing archival EVLA data. We find no radio emission from the AE Aur bow shock either. The corresponding upper limits for the absorbed (unabsorbed) X-ray flux of 5.9(7.8)x10^-15^erg/cm^2^/s (3{sigma}) and, in the radio range of 2mJy (1.4GHz) and 0.4mJy (5.0GHz), are used to put constraints on model predictions for particle acceleration within the bow shock. In the "classical" framework of diffusive shock acceleration, we find that the predicted X-ray and radio emission by the bow shock is at least two orders of magnitude below the current upper limits, consistent with the systematic nondetections of up to 60 stellar bow shocks. The only exception so far remains that of BD+43{deg}3654, which is probably the result of its very large mass-loss rate among runaway O stars.