We report on the coupling of a single nitrogen-vacancy (NV) center in a nanodiamond to a fiber-based microcavity at room temperature. Investigating the very same NV center inside the cavity and in free space allows us to systematically explore a regime of phonon-assisted cavity feeding. Making use of the NV center's strongly broadened emission, we realize a widely tunable, narrow band single photon source. A master equation model well reproduces our experimental results and predicts a transition into a Purcell-enhanced emission regime at low temperatures.