e present N-body simulations of resonant planets with inclined orbits that show chaotically evolving eccentricities and inclinations that can persist for at least 10 Gyr. A wide range of behavior is possible, from fast, low amplitude variations to systems in which eccentricities reach 0.9999 and inclinations 179.9 degrees. While the orbital elements evolve chaotically, at least one resonant argument always librates. We show that the HD 73526, HD 45364 and HD 60532 systems may be in chaotically-evolving resonances. Chaotic evolution is apparent in the 2:1, 3:1 and 3:2 resonances, and for planetary masses from lunar- to Jupiter-mass. In some cases, orbital disruption occurs after several Gyr, implying the mechanism is not rigorously stable, just long-lived relative to the main sequence lifetimes of solar-type stars. Planet-planet scattering appears to yield planets in inclined resonances that evolve chaotically in about 0.5% of cases. These results suggest that 1) approximate methods for identifying unstable orbital architectures may have limited applicability, 2) the observed close-in exoplanets may be produced during the high eccentricity phases induced by inclined resonances, 3) those exoplanets' orbital planes may be misaligned with the host star's spin axis, 4) systems with resonances may be systematically younger than those without, 5) the distribution of period ratios of adjacent planets detected via transit may be skewed due to inclined resonances, and 6) potentially habitable planets in resonances may have dramatically different climatic evolution than the Earth. The GAIA spacecraft is capable of discovering giant planets in these types of orbits.