Recent Internet measurements have found pervasive evidence of some surprising scaling properties. The two we focus on in this paper are self-similar scaling in the burst patterns of Internet traffic and, in some contexts, scale-free structure in the network's interconnection topology. These findings have led to a number of proposed models or "explanations" of such "emergent" phenomena. Many of these explanations invoke concepts such as fractals, chaos, or self-organized criticality, mainly because these concepts are closely associated with scale invariance and power laws. We examine these criticality-based explanations of self-similar scaling behavior---of both traffic flows through the Internet and the Internet's topology---to see whether they indeed explain the observed phenomena. To do so, we bring to bear a simple validation framework that aims at testing whether a proposed model is merely evocative, in that it can reproduce the phenomenon of interest but does not necessarily capture and incorporate the true underlying cause, or indeed explanatory, in that it also captures the causal mechanisms (why and how, in addition to what). We argue that the framework can provide a basis for developing a useful, consistent, and verifiable theory of large networks such as the Internet. Applying the framework, we find that, whereas the proposed criticality-based models are able to produce the observed "emergent" phenomena, they unfortunately fail as sound explanations of why such scaling behavior arises in the Internet.