Digital voltmeters generally display decimal digits. The resolution implied by that is not necessarily the same as the underlying binary A/D.
Since voltmeters for human display don't need to be fast, but high resolution is good, these usually use delta-sigma A/Ds inside. These can have 20 or more bits, but are "slow" relative to other A/D technologies. When you want to update a visible number no more than every 500 ms, even a "slow" 20 ms or 50 ms A/D is plenty fast.
However, what the voltmeter manufacturer promises is more than just resolution. Accuracy and linearity are important too. As a result, the internal A/D usually has significantly more resolution than is ultimately displayed, with some filtering between the raw values and the displayed value.
So, it's not so simple as a "4½ digit" voltmeter having a 16 bit A/D internally, although that is at least theoretically possible. However, it is very unlikely that the internal A/D steps match the decimal display steps. There is a processor between the internal A/D and the decimal display that can perform lots of functions, including applying some factory-measured calibration offset and scaling, filtering, range scaling, and then ultimate conversion to decimal.
Basically, the raw internal binary A/D result is a lot more decoupled from the displayed decimal value than you are imagining in your simplistic example.
some DMM is rated for 20000 counts, so it can only be 16 bits
It's not clear exactly what "20,000 counts" means. Almost certainly there is a binary A/D deep down in the meter. 20,000 is between 14 and 15 bits, so there is some conversion going on.
Since a 15 bit A/D is unlikely, all you can really say is that there is a 16 bit or more A/D inside. The "20,000 counts" spec may be the resolution after all conversions, range scaling, non-linearity, and other accuracy issues are taken into account. That actual A/D could be a 20 bit delta-sigma, for example.
Good spec sheets for binary A/Ds will show the number of effective bits after various sources of error are taken into account. For example, a 16 bit A/D produces 16 bits of result for each reading, but might only be good to 14½ bits under certain conditions. If you tilt your head just right, you could possibly describe that as 214.5 = 23,170, rounded to "20,000" counts.