A superconducting Coulomb-blockade electrometer was used to measure the Coulomb staircase of an Al/AlOx/Al Cooper-pair box from a temperature of 30 to 300 mK. At the lowest temperature, the Coulomb staircase displays effects from nonequilibrium quasiparticles. As the temperature is increased, an initial decrease is found in the width of the odd steps in the staircase, which corresponds to a reduction in the probability of having a quasiparticle on the island of the box. Above 180 mK, the width of the odd steps increases, eventually producing a staircase with 1e features. We develop a steady-state model of the system and find that the presence of quasiparticles at low temperature is consistent with the assumption of Aumentado [Phys. Rev. Lett. 92, 066802 (2004)] that nonequilibrium quasiparticles are generated in the leads. Above 180 mK, our results are consistent with the quasiparticle states of the island being thermally populated.