We report on simultaneous radio and X-ray observations of the repeating fast radio burst source FRB 180916.J0158+65 using the Canadian Hydrogen Intensity Mapping Experiment (CHIME), Effelsberg, and Deep Space Network (DSS-14 and DSS-63) radio telescopes and the Chandra X-ray Observatory. During 33 ks of Chandra observations, we detect no radio bursts in overlapping Effelsberg or Deep Space Network observations and a single radio burst during CHIME/FRB source transits. We detect no X-ray events in excess of the background during the Chandra observations. These non-detections imply a 5-$sigma$ limit of $<5times10^{-10}$ erg cm$^{-2}$ for the 0.5--10 keV fluence of prompt emission at the time of the radio burst and $1.3times10^{-9}$ erg cm$^{-2}$ at any time during the Chandra observations at the position of FRB 180916.J0158+65. Given the host-galaxy redshift of FRB 180916.J0158+65 ($zsim0.034$), these correspond to energy limits of $<1.6times10^{45}$ erg and $<4times10^{45}$ erg, respectively. We also place a 5-$sigma$ limit of $<8times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ on the 0.5--10,keV absorbed flux of a persistent source at the location of FRB 180916.J0158+65. This corresponds to a luminosity limit of $<2times10^{40}$ erg s$^{-1}$. Using Fermi/GBM data we search for prompt gamma-ray emission at the time of radio bursts from FRB 180916.J0158+65 and find no significant bursts, placing a limit of $4times10^{-9}$ erg cm$^{-2}$ on the 10--100 keV fluence. We also search Fermi/LAT data for periodic modulation of the gamma-ray brightness at the 16.35-day period of radio-burst activity and detect no significant modulation. We compare these deep limits to the predictions of various fast radio burst models, but conclude that similar X-ray constraints on a closer fast radio burst source would be needed to strongly constrain theory.
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