The US Food and Drug Administration (FDA) doesn't have much (that I can find) guidance for a threshold of dairy content above which it has to be listed on the label. There is, however, an article: What FDA Learned About Dark Chocolate and Milk Allergies (found in this Seasoned Advice question) which provides a summary of information from a larger FDA study: A Survey of Milk in Dark Chocolate Products.
Out of dark chocolate products that the tested, some had detectable traces of milk (> 2.5ppm). This includes vegan, lactose-free/vegan, and "dairy free" products, in addition to some which simply didn't list milk as an ingredient, but indicated they "may contain" traces of dairy.
One caveat of this study is that milk chocolate and dark chocolate are likely to be processed on the same equipment. If foods with dairy and vegan foodstuffs are processed on different machines, there is less possibility of cross-contamination. Therefore, I hesitate to extrapolate results for chocolate to all vegan products -- but based on the research, it seems clear that there is a strong chance of traces of dairy, eggs, or other animal-sourced ingredients appearing in processed foods intended to be vegan.
At that point, you get into ethical arguments and personal preference about whether these products are vegan (or "vegan enough"), which are covered fairly well in other answers :)