Optical emission from individual strained indium arsenide (InAs) islands buried in gallium arsenide (GaAs) was studied. At low excitation power density, the spectra from these quantum dots consist of a single line. At higher excitation power density, additional emission lines appeared at both higher and lower energies, separated from the main line by about 1 millielectron volt. At even higher excitation power density, this set of lines was replaced by a broad emission peaking below the original line. The splittings were an order of magnitude smaller than the lowest single-electron or single-hole excited state energies, indicating that the fine structure results from few-particle interactions in the dot. Calculations of few-particle effects give splittings of the observed magnitude.