Threshold photodetachment spectroscopy of the molecular ion C_{3}N^{-} has been performed at both 16(1) and 295(2) K in a 22-pole ion trap. The 295(2) K spectrum shows a large increase in the cross section with an onset about 200 cm^{-1} below threshold, which is explained by significant vibrational excitation of the trapped ions at room temperature. This excitation disappears at cryogenic temperatures leading to an almost steplike onset of the cross section at threshold, which cannot be adequately described with a Wigner threshold law. Instead, we show that the model developed by O'Malley for photodetachment from neutrals with large permanent dipoles [Phys. Rev. 137, A1668 (1965)PHRVAO0031-899X10.1103/PhysRev.137.A1668] fits very well to the data. A high-resolution scan of the threshold region yields additional features, which we assign to the rotational P and R branches of an electronic transition to a dipole-bound state with ^{1}Σ^{+} symmetry. This state is found 2(1) cm^{-1} below threshold in very good agreement with a recent computational prediction. We furthermore refine the value of the electron affinity of C_{3}N to be 34 727(1) cm^{-1}.