Core-collapse supernovae are a promising potential high-energy neutrino source class. We test for correlation between seven years of IceCube neutrino data and a catalog containing more than 1000 core-collapse supernovae of types IIn and IIP and a sample of stripped-envelope supernovae. We search both for neutrino emission from individual supernovae as well as for combined emission from the whole supernova sample, through a stacking analysis. No significant spatial or temporal correlation of neutrinos with the cataloged supernovae was found. All scenarios were tested against the background expectation and together yield an overall p-value of 93%; therefore, they show consistency with the background only. The derived upper limits on the total energy emitted in neutrinos are 1.7 x 10(48) erg for stripped-envelope supernovae, 2.8 x 10(48) erg for type IIP, and 1.3 x 10(49) erg for type IIn SNe, the latter disfavoring models with optimistic assumptions for neutrino production in interacting supernovae. We conclude that stripped-envelope supernovae and supernovae of type IIn do not contribute more than 14.6% and 33.9%, respectively, to the diffuse neutrino flux in the energy range of about [ 10(3)-10(5)] GeV, assuming that the neutrino energy spectrum follows a power-law with an index of -2.5. Under the same assumption, we can only constrain the contribution of type IIP SNe to no more than 59.9%. Thus, core-collapse supernovae of types IIn and stripped-envelope supernovae can both be ruled out as the dominant source of the diffuse neutrino flux under the given assumptions.