The magnetotail current layer is thought to be subject to a variety of instabilities. One instability arising from the presence of two ion populations, the cold lobe ions and the current-carrying hot plasma sheet ions, is the ion-ion kink mode. Detailed linear properties of this mode in the magnetotail were investigated by Karimabadi et al. <2003>, where it was shown that the mode differs from the standard Kelvin-Helmholtz instability. In this paper the nonlinear properties of the ion-ion kink mode are investigated using three-dimensional (3-D) full particle and hybrid (fluid electron, kinetic ions) simulations. It is shown that this mode is primarily driven by a velocity shear arising from the presence of multiple ion populations. The instability saturates as a result of broadening of the current layer and reduction of the velocity shear. The instability, however, differs in important aspects from the standard Kelvin-Helmholtz instability (KHI). Its linear mode properties exhibit dependencies on the kinetic details of the secondary ion population and its nonlinear evolution is found to be significantly different from previous MHD and Hall MHD treatments of the instability as well as from the KHI. In particular, the usual formation of vortices and coalescence that occur for the Kelvin-Helmholtz instability are absent for the ion-ion kink mode. Although the lobe ions can form vortices, such vortices are localized and do not affect current-carrying hot plasma sheet ions. Recent Cluster observations of modulated and bifurcated current sheets <Runov et al., 2003> are discussed within the context of the ion-ion kink mode. Hybrid simulations with open boundary conditions and using the parameters for this event demonstrate a very good agreement between the wavelength, period, and amplitude of the ion-ion kink mode and the observed wave-like disturbance. It is shown that the bifurcated current sheet can be explained in terms of a traveling kink displacement in which the current has a single continuous displacement into both hemispheres.