What is "linearity error"?
In order to make this easily understandable, some liberties are taken, so don't take this as being exactly how things really work. If you grasp the concept, that's all we're trying to achieve.
Let's take a test tone, 1 Khz.
It's a 16bit/44.1 KHz file, and we have two DACs, one with a 16-bit chip and one with a 24-bit chip.
Now, a DAC is supposed to produce a 2V output signal, some are a bit higher than that.
But that makes for some complicated math (dividing 2,000 mV by 16). So let's pretend they output 1600mV maximum instead, making each bit 100mV. Not to mention that there are far more than 16 discrete steps in level, but bear with me.
There are no DACs that are truly linear. In other words when asked to reproduce that 1KHz tone at a level equivalent to 2 bits (14bits down from full scale), it is supposed to output 200mV (using our example)
But no 16-bit DAC will do that. Generally linearity errors will begin to show up around the 13th bit down from full scale. So, instead of 200mV maybe it's 250mV or 140mV. And because it's non-linear, the 3-bit level might be only 220mV, or 330mV. The least significant bit is usually way off, plus or minus.
Now, take the same 16-bit file played back with a 24-bit DAC. Measurements reveal that it's linearity errors show up around the 19th or 20th bit down (typically). Which means it can perfectly play back all 16 bits without linearity error.
If we change the file to a 24-bit file, we are back to the same situation as with the 16-bit DAC; linearity error is inevitable.
Even though a 24-bit file is supposed to have more resolution than a 16-bit one, you may find that it doesn't sound that much different than a 16-bit version, because the actual resolution is much closer than at first it seems.
Now for a real-world example.
In the graph example, the response should be a straight line right to the -96dB point for a 16-bit file, and to the left edge of the graph for a 24-bit file. Although the error is a bit extreme (-8dB at -96dB) it does err on the side of "too quiet" and is at least consistent. Some skew wildly above and below ideal. We would say this DAC resolves down to about -68dB, which is equivalent to about 11~12 bits resolution.