We investigate the gas velocity dispersions of a sample of filaments recently detected as part of the Herschel Gould Belt Survey in the IC 5146, Aquila, and Polaris interstellar clouds. To measure these velocity dispersions, we use 13CO, C18O, and N2H+ line observations obtained with the IRAM 30 m telescope. Correlating our velocity dispersion measurements with the filament column densities derived from Herschel data, we show that interstellar filaments can be divided into two regimes: thermally subcritical filaments, which have transonic velocity dispersions (cs ≲ σtot < 2   cs) independent of column density and are gravitationally unbound; and thermally supercritical filaments, which have higher velocity dispersions scaling roughly as the square root of column density (σtot ∝ Σ00.5) and which are self-gravitating. The higher velocity dispersions of supercritical filaments may not directly arise from supersonic interstellar turbulence but may be driven by gravitational contraction/accretion. Based on our observational results, we propose an evolutionary scenario whereby supercritical filaments undergo gravitational contraction and increase in mass per unit length through accretion of background material, while remaining in rough virial balance. We further suggest that this accretion process allows supercritical filaments to keep their approximately constant inner widths (~0.1 pc) while contracting.