A/D and D/A conversion has come a long way in few short decades. The reputation that digital audio had early on for being cold and sterile, or even harsh, had nothing to do with the underlying theory and a lot to do with the implementation of conversion circuitry.
Digital clocks were one area where early converter designs often fell short. Filtering, and the propensity of analog antialiasing filtering to produce unwanted byproducts within the audio band, was another area where many designs were lacking. Clock accuracy, and the ability of devices to cleanly lock to external clocks, has improved by leaps and bounds, as has filter design, including the migration of filtering into the digital domain (thus concurrently migrating residual filtering artifacts out of the analog pass band) accommodated by over-sampling conversion schemes.
Bit depth is another area where we've made steady progress. Getting an honest 16- bit performance was difficult with early converter parts, and certainly expensive to pull off cost-effectively in a multitrack environment. As converter parts moved to 18-, 20-, and then 24-bit resolution, the performance improved simultaneously. On one occasion, when high-quality 20-bit converters were just becoming the state of the art, I asked audio guru, George Massenburg, what bit depth resolution he'd consider acceptable. "Twenty-and-a-half bits" was his answer. I liked this answer a lot: 19-20-bits for audio, for a dynamic range of about 114-120 dB, plus a half-bit to a bit-and-a-half for dither. I don't know if George would give that same answer today, but with current 24- bit resolution, even on relatively inexpensive gear, that target is fairly routinely realized. A true 24-bit conversion is still out of reach, partly because the laws of physics impose some practical limitations; we're getting damn close, though.
The early converter chips did little more than convert, while today, all of the surrounding circuitry once necessary to implement a converter — sample and hold, filtering, clock recovery, external interface support, and so on — is built into the chips. If each of these elements is not implemented well inside a chip, performance can suffer, but the chips do just keep getting better. There are designers whose expertise is in finessing the small details, wringing the most out of a given chipset, but even a designer of average skill can achieve good, if not great, performance.
While the digital bits are important, let's not forget that analog circuitry precedes conversion to digital, and follows conversion from digital. All of the traditional elements of analog circuit design like slew rate, dynamic range, linearity, and even the cleanliness and reserve of the power supply rails come into play. Merely adequate analog design can compromise the best digital performance — converter quality is the sum total of the analog and digital performance of the signal path.
Given attention to detail and care in implementation, modern digital conversion can be stellar, transparent to most ears. A bit here, a bit there — pretty soon we're talking serious audio!