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We report about the effect of nitrogen and carbon concentration on the superconducting transition temperature TC of (NbMoTaW)1CxNy carbonitride films deposited using reactive DC magnetron sputtering. By measuring the temperature dependence of electrical resistance and magnetization of these carbonitrides, with 0.20 x 1.17 and 0 y 0.73, we observe a TC enhancement that occurs especially at high (x 0.76) carbon concentrations, with the largest TC = 9.6 K observed in the over-doped fcc crystal structure with x = 1.17 and y = 0.41. The reason why the largest TC appears at high carbon concentrations is probably related to the lower atomic mass of carbon compared to nitrogen and to the increase of the electron-phonon interaction due to different bonding of carbon (compared to nitrogen) to the Nb-Mo-Ta-W metallic sublattice. However, as for concentrations y 0.71 and x + y 1.58 two structural phases begin to form, also the proximity to structural instability may play a role in the observed TC enhancement. Additional measurements in magnetic field show that the upper critical fields Bc2 of (NbMoTaW)1CxNy carbonitrides provide Bc2/TC 2 T/K ratios, which are within the weak-coupling pair breaking limit.