We report on the quadrupling of the transition spectrum of an Al/AlOx/Al Cooper-pair box (CPB) charge qubit in the 4.0-7.3 GHz frequency range. The qubit was coupled to a quasilumped-element Al superconducting resonator and measured at a temperature of 25 mK. We obtained good matches between the observed spectrum and the spectra calculated from a model Hamiltonian containing two distinct low-excitation energy two-level systems (TLS) coupled to the CPB. In our model, each TLS has a charge that tunnels between two sites in a local potential and induces a change in the CPB critical current. By fitting the model to the spectrum, we have extracted microscopic parameters of the fluctuators including the well asymmetry, tunneling rate, and a surprisingly large fractional change (30%-40%) in the critical current (12 nA). This large change is consistent with a Josephson junction with a nonuniform tunnel barrier containing a few dominant conduction channels and a TLS that modulates one of them.