Many disordered superconducting films exhibit smeared tunneling spectra with evident in-gap states. We have found that the tunneling density of states in ultrathin MoC films is gapless and can be described by the Dynes version of the BCS density of states with a strong broadening parameter Gamma accounting for the suppression of coherence peaks and increased in-gap states. The thinner the film, the lower the T-c and the superconducting energy gap Delta and the larger the Gamma. MoC films of 3 nm thickness deposited simultaneously on silicon and sapphire substrates reveal very similar scalar disorder, evidenced by the equal sheet resistance, but exhibit different superconducting characteristics of T-c, Delta and Gamma, suggesting that pair breaking responsible for the dissipation channel and the suppression of superconductivity originates on the film-substrate interface. It indicates that sapphire is a stronger pair breaker. Interface pair breaking can be operative in other cases as well.