We report a joint experimental and theoretical study of the interference properties of a single-photon source based on a In(Ga)As quantum dot embedded in a quasiplanar GaAs microcavity. Using resonant laser excitation with a pulse separation of 2 ns, we find near-perfect interference of the emitted photons, and a corresponding indistinguishability of $\mathcal{I}=(99.6{\phantom{\rule{0.16em}{0ex}}}_{\ensuremath{-}\phantom{\rule{0.16em}{0ex}}1.4}^{+\phantom{\rule{0.16em}{0ex}}0.4})%$. For larger pulse separations, quasiresonant excitation conditions, increasing pump power, or with increasing temperature, the interference contrast is progressively and notably reduced. We present a systematic study of the relevant dephasing mechanisms and explain our results in the framework of a microscopic model of our system. For strictly resonant excitation, we show that photon indistinguishability is independent of pump power, but strongly influenced by virtual phonon-assisted processes which are not evident in excitonic Rabi oscillations.