Basic Question: Do the available Internet-related connectivity measurements and their analysis support the sort of claims that can be found in the existing complex networks literature? Key Issues: What about data hygiene? What about statistical rigor? What about model validation? Author discusses some of the main problems and challenges associated with measuring, inferring, and modeling various types of Internet-related connectivity structures. To this end, he uses some known examples to illustrate the need to understand the process by which Internet connectivity measurements are obtained, explore the sensitivity of inferred graph properties to known ambiguities in the data, be more critical with respect to the dominant, preferential attachmenttype network modeling paradigm, and be more serious/ambitious when it comes to model validation. Ignoring any of these issues is bound to produce results that are best described by the well-known aphorism "lies, damned lies, and statistics."