  We report EGRET upper limits on the high-energy gamma-ray emission from
clusters of galaxies. EGRET observations between 1991 and 2000 were analyzed at
positions of 58 individual clusters from a flux-limited sample of nearby X-ray
bright galaxy clusters. Subsequently, a coadded image from individual galaxy
clusters has been analyzed using an adequately adapted diffuse gamma-ray
foreground model. The resulting 2 sigma upper limit for the average cluster is
\~ 6 x 10^{-9} cm^{-2} s^{-1} for E > 100 MeV. Implications of the
non--detection of prominent individual clusters and of the general inability to
detect the X-ray brightest galaxy clusters as a class of gamma-ray emitters are
discussed. We compare our results with model predictions on the high-energy
gamma-ray emission from galaxy clusters as well as with recent claims of an
association between unidentified or unresolved gamma-ray sources and Abell
clusters of galaxies and find these contradictory.
