Max useful filter tap count for given fixed point bitwidth
Since you can widen your bit width with each operation, no, there's no such maximum useful width.

Take GPS receivers as an example: many of them citation needed use 2 bit ADCs and yet are useful, because the signal, already hidden in noise before it reaches the ADC, only gets "visible" by massive processing gain.

A typical step with massive processing gain would be a very sharp digital filter. Those can get very long. 

There's a lot of experienced audio and video folks on here. They can tell you much more about the astronomical lengths high-quality filters can get.