In this paper we continue the study of geometric/asymptotic properties of adaptive nonlinear systems. The long-standing question of whether the parameter estimates converge to stabilizing values-stabilizing if used in a nonadaptive controller-is addressed in the general set-point regulation case. The key quantifier of excitation in an adaptive system is the rank r of the regressor matrix at the resulting equilibrium. Our earlier paper showed that when either r = 0 or r = p (where p is the number of uncertain parameters), the set of initial conditions leading to destabilizing estimates is of measure zero. Intuition suggests the same for the intermediate case 0 < r < p studied in this paper. We present a surprising result : the set of initial conditions leading to destabilizing estimates can have positive measure. We present results for the backstepping design with tuning functions; the same results can be established for other Lyapunov-based adaptive designs.