  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
$<5\times10^{-10}$ erg cm$^{-2}$ for the 0.5--10 keV fluence of prompt emission
at the time of the radio burst and $1.3\times10^{-9}$ 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 ($z\sim0.034$), these
correspond to energy limits of $<1.6\times10^{45}$ erg and $<4\times10^{45}$
erg, respectively. We also place a 5-$\sigma$ limit of $<8\times10^{-15}$ 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
$<2\times10^{40}$ 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 $4\times10^{-9}$ 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.
