Intense magnetic storms often develop in two stages such that a second ring current enhancement begins before the first ring current enhancement has recovered to the prestorm level. Since Dst traces of such storms exhibit two dips, we refer to these as double-dip storms. Here we compare double- and single-dip storms with similar convective and diffusive transport characteristics for effectiveness at forming the proton ring current. Our model storms consist of superposition of almost randomly occurring impulses in the convection electric field. We have synthesized a hypothetical double-dip storm consisting of a moderate 6-hour storm, followed by a 3-hour quiet interval and then by a more intense 15-hour storm, for a total duration of 24 hours. For comparison, we consider a single-dip model storm with an unmodulated 24-hour main phase during which the root-mean-square enhancement of the cross-tail potential drop is made equal to the time-weighted rms enhancement for the double-dip model storm. This leads to comparable time-averaged diffusion coefficients for our single- and double-dip model storms. The mean enhancement of the cross-tail potential drop of the two storms are also comparable. When the stormtime proton plasma sheet distribution, the source of ring current protons, is left unchanged from its quiet (prestorm) level, we find little difference in proton energy content per unit R (which is geocentric distance normalized by RE) between our double- and single-dip model storms. The proton-energy content of the magnetosphere is roughly increased by a factor of 2.5 by either model storm under this scenario, in which the overall amount of stormtime transport, whether convective or diffusive, is nearly the same for the double- and single-dip model storms. As in our earlier work, we require here an enhanced stormtime plasma sheet population (in addition to enhanced particle transport) in order to achieve (for example) the 20-fold increase in |Dst| characteristic of a large storm. Only when we invoke a two-stage stormtime enhancement of the boundary (plasma sheet) phase space density in combination with the two-stage enhancement in particle transport, our double-dip model storm does show a much larger total energy content than our single-dip model storm with a one-stage enhancement of the boundary spectrum. This suggests that plasma sheet preconditioning may be important for the development of especially intense storms. ¿ 2000 American Geophysical Union