We have observed a few distinct anomalous avoided level crossings and voltage-dependent transitions in the excited-state spectrum of an Al/AlOx/Al Cooper-pair box (CPB). The device was measured at 40 mK in the 15-50 GHz frequency range. We find that a given level crosses the CPB spectrum at two different gate voltages; the frequency and splitting size of the two crossings differ and the splitting size depends on the Josephson energy of the CPB. We show that this behavior is not only consistent with the CPB being coupled to discrete charged "two-level" quantum systems, which move atomic distances in the CPB junctions, but that the spectra provide information about the fluctuators, which is not available from phase qubit spectra of anomalous avoided levels. In particular by fitting a model Hamiltonian to our data, we extract microscopic parameters for each fluctuator, including well asymmetry, tunneling amplitude, and the minimum hopping distance for each fluctuator. The tunneling rates range from less than 3.5-13 GHz, which represent values between 5 and 150% of the well asymmetry, and the dipole moments give a minimum hopping distance of 0.3-0.8 A. We have also found that these discrete two-level systems have a pronounced effect on the relaxation time (T-1) of the quantum states of the CPB and hence can be a source of dissipation for superconducting quantum bits.