A DAC is unlikely to generate additional distortion on its own only because its volume is too high, as it is (usually) designed to obtain a unit gain corresponding to the maximum level of digital signal at its input. If a 16-bit DAC maxes out its analog output when the input signal is - say - 2^14, then it means that it is effectively operating on a dynamic range of 14-bit, and should be advertised as such. Of course things are not so straightforward, as even the best DACs are not perfectly linear, but they definitely should not clip the output signal.