FRB 180916 is a most intriguing source capable of producing repeating fast radio bursts with a periodic 16.3 day temporal pattern. The source is well positioned in a star-forming region in the outskirts of a nearby galaxy at 150 Mpc distance. In this Letter we report on the X-ray and γ-ray observations of FRB 180916 obtained by AGILE and Swift. We focused especially on the recurrent 5 day time intervals of enhanced radio bursting. In particular, we report on the results obtained in the time intervals 2020 February 3-8, 2020 February 25, 2020 March 5-10, and 2020 March 22-28 during a multiwavelength campaign involving high-energy and radio observations of FRB 180916. We also searched for temporal coincidences at millisecond timescales between the 32 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 γ-ray emission on timescales of hours/days/weeks. Our cumulative X-ray (0.3-10 keV) flux upper limit of 5 × 10−14 erg cm−2 s−1 (obtained during 5 day active intervals from several 1-2 ks integrations) translates into an isotropic luminosity upper limit of LX,UL ∼ 1.5 × 1041 erg s−1. Deep γ-ray 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 upper limits on high-energy emission from the FRB 180916 source. Our results constrain the dissipation of magnetic energy from a magnetar-like source of radius Rm, internal magnetic field Bm, and dissipation timescale τd to satisfy the relation , where Rm,6 is Rm in units of 106 cm, Bm,16 is Bm in units of 1016 G, and τd,8 in units of 108 s.