A search has been made for neutrinos from the hep reaction in the Sun and from the diffuse supernova neutrino background (DSNB) using data collected during the first operational phase of the Sudbury Neutrino Observatory, with an exposure of 0.65 kilotonne-years. For the hep neutrino search, two events are observed in the effective electron energy range of 14.3 MeV < Teff < 20 MeV where 3.1 background events are expected. After accounting for neutrino oscillations, an upper limit of 2.3 x 104 cm-2s-1 at the 90 percent confidence level is inferred on the integral total flux of hep neutrinos. For DSNB neutrinos, no events are observed in the effective electron energy range of 21 MeV < Teff < 35 MeV and, consequently, an upper limit on the nu e component of the DSNB fluxin the neutrino energy range of 22.9 MeV < E nu < 36.9 MeV of 70 cm-2-1 is inferred at the 90 percent confidence level. This is an improvement by a factor of 6.5 on the previous best upper limit on the hep neutrino flux and by two orders of magnitude on the previous upper limit on the nu e component of the DSNB flux.