We describe how the nonlinear development of the R-mode instability of neutron stars influences spin up to millisecond periods via accretion. When nearly resonant interactions of the l = m = 2 R-mode with pairs of "daughter modes" are included, the R-mode saturates at the lowest amplitude which leads to significant excitation of a pair of modes. The lower bound for this threshold amplitude is proportional to the damping rate of the particular daughter modes that are excited parametrically. We show that if dissipation occurs in a very thin boundary layer at the crust-core boundary, the R-mode saturation amplitude is too large for angular momentum gain from accretion to overcome loss to gravitational radiation. We find that lower dissipation is required to explain spin up to frequencies much higher than 300 Hz. We conjecture that if the transition from the fluid core to the crystalline crust occurs over a distance much longer than 1 cm, then a sharp viscous boundary layer fails to form. In this case, damping is due to shear viscosity dissipation integrated over the entire star. We estimate the lowest parametric instability threshold from first principles. The resulting saturation amplitude is low enough to permit spin up to higher frequencies. The requirement to allow continued spin up imposes an upper bound to the frequencies attained via accretion that plausibly may be about 750 Hz. Within this framework, the R-mode is unstable for all millisecond pulsars, whether accreting or not.