<p>I am re-writing this question because, well, it needs to be updated.</p>

<p>I have the Anet A6, but in a general sense of things, what kind of threads can I produce before it no longer works? </p>

This depends on the nozzle diameter, the layer thickness, and the material.
I've made very good M8 and acceptable M6 threads (nut and bolt) at 0.2mm layers with a 0.5mm nozzle, out of ABS, and also out of PETG.