Broadband noise on supercontinuum spectra generated in microstructure fiber is shown to lead to amplitude fluctuations as large as 50 % for certain input laser pulse parameters. We study this noise using both experimental measurements and numerical simulations with a generalized stochastic nonlinear Schroedinger equation, finding good quantitative agreement over a range of input pulse energies and chirp values. This noise is shown to arise from nonlinear amplification of two quantum noise inputs: the input pulse shot noise and the spontaneous Raman scattering down the fiber.