Mean motion resonances, in which two orbital frequencies are close to an integer multiple of each other, are common throughout the Solar System and exoplanetary systems. We present N-body simulations of resonant planets with inclined orbits and show that orbital eccentricities and inclinations can evolve chaotically for at least 10 Gyr. A wide range of behavior is possible, ranging from fast, low amplitude variations to a complete sampling of all parameter space, i.e. eccentricities reach 0.999 and inclinations 179.9 degrees. While the orbital elements evolve chaotically, at least one resonant argument librates, the traditional metric for identifying resonant behavior. This chaotic evolution is possible in the 2:1, 3:1 and 3:2 resonances, and for a range of 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. We also re-examine simulations of planet-planet scattering and find that they produce planets in inclined resonances that evolve chaotically in about 0.5% of cases. Our results suggest that 1) approximate methods for identifying unstable orbital architectures may have limited applicability, 2) some short-period exoplanets may be formed during tidal circularization when the eccentricity is large, 3) those exoplanets' orbital planes may be misaligned with the host star spin axis, 4) on average, 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, and 6) potentially habitable planets may have dramatically different climatic evolution than the Earth. We show that the known systems HD 73526, HD 45364 and HD 60532 system may be in chaotically-evolving resonances. The GAIA spacecraft is capable of discovering giant planets in these types of planetary systems.