The cross sections for two-neutron removal from 15,16C on a carbon target have been measured at around 240 MeV/nucleon. The measured cross section for 15C is smaller than for 16C. The trends of the cross sections for 15,16C are inconsistent with the previously reported experimental data, but in agreement with the theoretical predictions based on eikonal-model calculations. Based on the present results of 15,16C combined with the available experimental data on 17–20C, an odd-even staggering of the two-neutron removal cross sections along the neutron-rich carbon isotopic chain from 15C to 20C is observed, and this feature is well reproduced by the theoretical calculations.