Atomic motion of guest atoms inside semiconducting clathrate cages is considered as an important source for the glasslike thermal behavior. 69 Ga and 71 Ga Nuclear Magnetic Resonance (NMR) studies on type-I Ba8Ga16Sn30 show a clear low temperature relaxation peak attributed to the influence of Ba rattling dynamics on the framework-atom resonance, with a quadrupolar relaxation mechanism as the leading contribution. The data are analyzed using a two-phonon Raman process, according to a recent theory involving localized anharmonic oscillators. Excellent agreement is obtained using this model, with the parameters corresponding to a uniform array of localized oscillators with very large anharmonicity.