We report measurements of Rabi oscillations and spectroscopic coherence times in an Al/AlO(x)/Al and three Nb/AlO(x)/Nb dc SQUID phase qubits. One junction of the SQUID acts as a phase qubit and the other junction acts as a current-controlled nonlinear isolating inductor, allowing us to change the coupling to the current bias leads in situ by an order of magnitude. We found that for the Al qubit, a spectroscopic coherence time T(2)(*) varied from 3 to 7 ns and the decay envelope of Rabi oscillations had a time constant T =25 ns on average at 80 mK. The three Nb devices also showed T(2)(*) in the range of 3-8 ns, but T was 9-15 ns, just about 1/2 the value we found in the Al device. For all the devices, the time constants were roughly independent of the isolation from the bias lines, implying that noise and dissipation from the bias leads were not the principal sources of dephasing and inhomogeneous broadening.