Abstract: This paper examines contrasting notions of self-organization in the Internet and next-generation communication networks, by reviewing in some detail recent evidence regarding several of the more popular attempts to explain prominent features of Internet structure and behavior as ``emergent phenomena''. In these examples, what might appear to the non-expert as ``emergent self-organization'' in the Internet actually results from well-conceived (albeit perhaps ad hoc) design, with explanations that are mathematically rigorous, in agreement with engineering reality, and fully consistent with network measurements. These examples serve as concrete starting points from which networking researchers can assess whether or not explanations involving self-organization are relevant or appropriate in the context of next-generation communication networks, while also highlighting the main differences between approaches to self-organization that are rooted in engineering design versus those inspired by statistical physics.