  FRB 180916 is a most intriguing source at 150 Mpc distance capable of
producing repeating fast radio bursts with a periodic 16.35 day temporal
pattern. We report on the X-ray and $\gamma$-ray observations of FRB 180916
obtained by AGILE and Swift. We focused on the recurrent 5-day time intervals
of active radio bursting and present results obtained on Feb. 3 - 8; Feb. 25;
Mar. 5 - 10; Mar. 22 - 28, 2020 during a multiwavelength campaign involving
high-energy and radio observations. We also searched for temporal coincidences
at millisecond timescales between all known radio bursts of FRB 180916 and
X-ray and MeV events detectable by AGILE. We do not detect any simultaneous
event or any extended X-ray and $\gamma$-ray emission on timescales of
hours/days/weeks. Our cumulative X-ray (0.3-10 keV) flux upper limit of $5
\times\,10^{-14} \rm \, erg \, cm^{-2} s^{-1}$ (obtained during 5-day active
intervals) translates into an isotropic luminosity upper limit of $L_{X,UL}
\sim 1.5 \times\, 10^{41} \rm erg \, s^{-1}$. Observations above 100 MeV over a
many-year timescale provide an average luminosity upper limit one order of
magnitude larger. These results provide the so-far most stringent limits on
high-energy emission from FRB 180916 and constrain the dissipation of magnetic
energy from a magnetar-like source of radius $R_m$, internal magnetic field
$B_m$ and dissipation timescale $\tau_d$ to satisfy the relation $R_{m,6}^3
B_{m,16}^2 \tau_{d,8}^{-1} \lesssim 1$, where $R_{m,6}$ is $R_m$ in units of
$10^6$ cm, $B_{m,16}$ is $B_m$ in units of $10^{16}$ G, and $\tau_{d,8}$ in
units of $10^8$ s.
