High-energy neutrino emission has been predicted for several short-lived astrophysical transients including gamma-ray bursts (GRBs), core-collapse supernovae with choked jets, and neutron star mergers. IceCube's optical and x-ray follow-up program searches for such transient sources by looking for two or more muon neutrino candidates in directional coincidence and arriving within 100 s. The measured rate of neutrino alerts is consistent with the expected rate of chance coincidences of atmospheric background events and no likely electromagnetic counterparts have been identified in Swift follow-up observations. Here, we calculate generic bounds on the neutrino flux of short-lived transient sources. Assuming an E^{-2.5} neutrino spectrum, we find that the neutrino flux of rare sources, like long gamma-ray bursts, is constrained to <5% of the detected astrophysical flux and the energy released in neutrinos (100 GeV to 10 PeV) by a median bright GRB-like source is <10^{52.5} erg. For a harder E^{-2.13} neutrino spectrum up to 30% of the flux could be produced by GRBs and the allowed median source energy is <10^{52} erg. A hypothetical population of transient sources has to be more common than 10^{-5} Mpc^{-3} yr^{-1} (5×10^{-8} Mpc^{-3} yr^{-1} for the E^{-2.13} spectrum) to account for the complete astrophysical neutrino flux.