The DAT recording medium was 16 bits wide. Period. [see later reply below for an amendment to this statement ... --ds] By the time DAT was invented, all the record logic was in ICs that couldn't be modified by mortal beings. I am quite sure that this person was recording no more than 16 bits per sample.
Noise-shaping can reduce the APPARENT noise level of a 16-bit channel to some extent, but this is done by shifting noise upwards in frequency to where we are somewhat less sensitive to it--basically sweeping it under the rug. Given any fixed number of bits to work with, you can't increase the actual dynamic range across the entire frequency band; you can only rob Peter to pay Paul. But there fairly soon comes a point--which differs with different approaches and different listeners--past which any further shifting of the noise causes a tangible feeling of "something's not right"-ness about the sound.
Sony, with their "Super Bit-Mapping" approach, found that point to be about 10 dB. In retrospect, I think a lot of engineers wish that they could have limited themselves to, say, 6 dB instead. They had a professional and a consumer version of the scheme; the SBM-1 adapter for portable DAT recorders was of course the consumer-grade system. But even the professional system has a rather bad reputation in retrospect--I never worked with it personally, but the main complaint I've heard from other engineers was audible artifacts at very low signal levels, and of course once you've applied this processing, it's "baked in" to the signal and can't be undone. Other vendors' noise-shaping approaches seem to have held up better--Apogee as I recall was said to have a good one. But best of all is not to try to fool people's hearing, and to use straight linear PCM with a channel quiet enough to accommodate your program material.
--Back when the Sony PCM-F1 was a new thing, I used to record concerts in its 14-bit mode, which had better dropout protection but inadequate dithering. Since at that time there was no way to edit consumer digital recordings or to transfer their contents digitally to any other system, I knew that the next step was always going to be analog dubbing. So I used Dolby "A"-type noise reduction with the PCM-F1, which gave it nearly the dynamic range of 16-bit recording along with the increased dropout protection. There were even more powerful analog noise reduction systems (Dolby SR, telcom c4) which could have extended that range beyond the raw 16-bit level to some extent.
--best regards