In the interstellar medium (ISM) ion-molecule reactions play a key role in forming complex molecules. Since 2006, after the radioastronomical discovery of the first of by now six interstellar anions, interest has grown in understanding the formation and destruction pathways of negative ions in the ISM. Experiments have focused on reactions and photodetachment of the identified negatively charged ions. Hints were found that the reactions of CnH- with H-2 may proceed with a low (<10(-13) cm(3) s(-1)), but finite rate [Eichelberger, B.; et al. Astrophys. J. 2007, 667, 1283]. Because of the high abundance of molecular hydrogen in the ISM, a precise knowledge of the reaction rate is needed for a better understanding of the low-temperature chemistry in the ISM. A suitable tool to analyze rare reactions is the 22-pole radiofrequency ion trap. Here, we report on reaction rates for C-n(-) and CnH- (n = 2, 4, 6) with buffer gas temperatures of H-2 at 12 and 300 K. Our experiments show the absence of these reactions with an upper limit to the rate coefficients between 4 X 10(-16) and 5 X 10(-15) cm(3) s(-1), except for the case of C-2(-), which does react with a finite rate with H-2 at low temperatures. For the cases of C2H- and C4H-, the experimental results were confirmed with quantum chemical calculations. In addition, the possible influence of a residual reactivity on the abundance of C4H- and C6H- in the ISM were estimated on the basis of a gas-phase chemical model based on the KIDA database. We found that the simulated ion abundances are already unaffected if reaction rate coefficients with H-2 were below 10(-14) cm(3) s(-1).