We study decoherence effects in qubits coupled to environments that exhibit resonant frequencies in their spectral function. We model the coupling of the qubit to its environment via the Caldeira-Leggett formulation of quantum dissipation/coherence, and study the simplest example of decoherence effects in circuits with resonances such as a dc superconducting quantum interference device (SQUID) phase qubit in the presence of an isolation circuit. Using the dc SQUID phase qubit as an example, we obtain two quite general results. First, when the frequency of the qubit is at least two times larger than the resonance frequency of the environmental spectral density (the isolation circuit in the case of a dc SQUID phase qubit), we find that the decoherence time of the qubit is a few orders of magnitude larger than that of the typical Ohmic regime, where the frequency of the qubit is much smaller than the resonance frequency of the spectral density. Second, we show that, when the qubit frequency is nearly the same as the resonant frequency of the environmental spectral density, an oscillatory non-Markovian decay emerges, as the qubit and its environment self-generate Rabi oscillations of characteristic time scales shorter than the decoherence time.