Surprisingly, yes. With noise shaping (https://en.wikipedia.org/wiki/Noise_shaping), very coarsely-quantized digital audio can produce high signal to noise ratios in the audible frequencies, via quantization techniques that push the error towards ultrasonic frequencies.
This doesn't violate information-theoretic limits because noise shaping requires very high sampling rates. The 1-bit Sony DSD format (https://en.wikipedia.org/wiki/Direct_Stream_Digital) used a 2.8MHz sample rate.
In the case of vinyl, the effective sample rate is physically limited by the (linear) record speed divided by the vinyl grain size, and to a rough approximation the bit depth would be log of the maximum groove amplitude divided by the grain size. However, the analog cutting mechanism would greatly limit the opportunity for dithering and noise shaping -- for example a needle cannot cut a wave shorter than the tip size.