We undertook coordinated campaigns with the Green Bank, Effelsberg, and Arecibo radio telescopes during Chandra X-ray Observatory and XMM-Newton observations of the repeating fast radio burst FRB 121102 to search for simultaneous radio and X-ray bursts. We find 12 radio bursts from FRB 121102 during 70 ks total of X-ray observations. We detect no X-ray photons at the times of radio bursts from FRB 121102 and further detect no X-ray bursts above the measured background at any time. We place a 5$sigma$ upper limit of $3times10^{-11}$ erg cm$^{-2}$ on the 0.5--10 keV fluence for X-ray bursts at the time of radio bursts for durations $<700$ ms, which corresponds to a burst energy of $4times10^{45}$ erg at the measured distance of FRB 121102. We also place limits on the 0.5--10 keV fluence of $5times10^{-10}$ erg cm$^{-2}$ and $1times10^{-9}$ erg cm$^{-2}$ for bursts emitted at any time during the XMM-Newton and Chandra observations, respectively, assuming a typical X-ray burst duration of 5 ms. We analyze data from the Fermi Gamma-ray Space Telescope Gamma-ray Burst Monitor and place a 5$sigma$ upper limit on the 10--100 keV fluence of $4times10^{-9}$ erg cm$^{-2}$ ($5times10^{47}$ erg at the distance of FRB 121102) for gamma-ray bursts at the time of radio bursts. We also present a deep search for a persistent X-ray source using all of the X-ray observations taken to date and place a 5$sigma$ upper limit on the 0.5--10 keV flux of $4times10^{-15}$ erg s$^{-1}$ cm$^{-2}$ ($3times10^{41}$ erg~s$^{-1}$ at the distance of FRB 121102). We discuss these non-detections in the context of the host environment of FRB 121102 and of possible sources of fast radio bursts in general.
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