  We perform X-ray studies of the shell-type supernova remnant (SNR)
gamma-Cygni associated with the brightest EGRET unidentified source 3EG
J2020+4017. In addition to the thermal emissions with characteristic
temperature of kT = 0.5-0.9 keV, we found an extremely hard X-ray component
from several clumps localized in the northern part of the remnant. This
component is described by a power-law with a photon index of 0.8-1.5. Both the
absolute flux and the spectral shape of the nonthermal X-rays cannot be
explained by the synchrotron or inverse-Compton mechanisms. We argue that the
unusually hard X-ray spectrum can be naturally interpreted in terms of
nonthermal bremsstrahlung from Coulomb-loss-flattened electron distribution in
dense environs with the gas density about 10 to 100 cm^-3 . For given spectrum
of the electron population, the ratio of the bremsstrahlung X- and gamma-ray
fluxes depends on the position of the ``Coulomb break'' in the electron
spectrum. The bulk of gamma-rays detected by EGRET would come from the
radio-bright and X-ray dim cloud at southeast, where very dense gas and strong
magnetic field would illuminate the cloud in the radio and gamma-ray bands, but
suppress the bremsstrahlung X-ray emission due to the shift of the ``Coulomb
break'' in the electron spectrum towards higher energies.
