Power-law distributions are an important quantity often used to characterize complex networks. Recent statistical investigations, however, question their universality, suggesting that scale-freeness is heterogeneous; particularly, the fitness of power-law distribution is weaker in social networks than in technological and biological networks. Yet less is known about the mechanisms behind this non-scale-free finding. We develop an activity-constraint approach where the finite tie-completing capacity of hubs or high-quality actors conditions the emergence of the heavy tail. We first construct synthetic models to examine how degree distribution behaves differently depending on activity constraints. Then we leverage the new data of a hip-hop collaboration network—a context where artists typically face an upper bound in accepting “featuring” invites on other artists’ songs—and find a sublinear relationship between artists’ quality and featuring indegree. In the featuring network that exhibits a better fit to the log-normal distribution, high-quality artists do receive more ties but not at the level expected by their quality. Work searching for empirical scale-free networks should differentiate tie-formation types since activity constraints as a mechanism, in part, determine finite or infinite growth, which then moderates how its degree distribution approaches a power law or other alternatives.