  We study the fluctuations of an energy level as a function of the number of
electrons $m$ added to a Coulomb-blockade quantum dot. A microscopic
calculation in the limit of Koopmans' theorem predicts that the standard
deviation of these fluctuations behaves as $\sqrt{m}$ in the absence of surface
charge but is linear in $m$ when the effect of a surface charge in a finite
geometry is included. The microscopic results are compared to a parametric
random-matrix approach. We estimate the number of electrons it takes to
scramble the spectrum completely in terms of the interaction strength, the
dimensionless Thouless conductance, and the symmetry class.
