This paper discusses two NISs: the public telephone network (PTN) and the Internet. Being themselves large and complex NISs, they not only merit study in their own right but can help us to understand some of the technical problems faced by the developers and operators of other NISs. In addition, the high cost of building a global communications infrastructure from the ground up implies that one or both of these two networks is likely to furnish communications services for most other NISs. Therefore, an understanding of the vulnerabilties of the PTN and Internet informs the assessment of the trustworthiness of other NISs. Ideas for improving the trustworthiness of the PTN and Internet are also proposed, both for the short-term (by improved use of existing technologies and procedures) and for the long-term (by identifying some areas where the state-of-the-art is inadequate and research is therefore needed). Finally, some observations are offered about Internet telephony and the use of the Internet for critical infrastructures.