If supernova remnants (SNRs) are the site of cosmic-ray acceleration, the associated nuclear interactions should result in observable fluxes of TeV gamma-rays from the nearest SNRs. Measurements of the gamma-ray flux from six nearby, radio bright, SNRs have been made with the Whipple gamma-ray telescope. No significant emission has been detected and upper limits on the >300 GeV flux are reported. Three of these SNRs (IC443, gamma-Cygni and W44) are spatially coincident with low latitude unidentified sources detected with EGRET. If the EGRET gamma-ray fluxes result from cosmic-ray interactions, then the EGRET and Whipple data are found to be collectively inconsistent with a cosmic-ray source flatter than E-2.4. These data weaken the case for the simplest models of shock acceleration and energy dependent propagation.