Recent studies concerning the Internet connectivity at the AS level have attracted considerable attention. These studies have exclusively relied on the BGP data from Oregon route-views [1] to derive some unexpected and intriguing results. The Oregon route-views data sets reflect AS peering relationships, as reported by BGP, seen from a handful of vantage points in the global Internet. The possibility that these data sets from Oregon route-views may provide only a very sketchy picture of the complete inter-AS connections that exist in the actual Internet has received surprisingly little scrutiny. In this paper, we will use the term "AS peering relationship" to mean that there is "at least one direct router-level connection" between two existing ASs, and that these two ASs agree to exchange traffic by enabling BGP between them. By augmenting the Oregon route-views data sets with BGP summary information from a large number of Internet Looking Glass sites and with routing policy information from Internet Routing Registry (IRR) databases, we find that (1) a significant number of existing AS connections remain hidden from most BGP routing tables, (2) the AS connections to tier-1 ASs are in general more easily observed than those to non tier-1 ASs, and (3) there are at least about 25--50% more AS connections in the Internet than commonly-used BGP-derived AS maps reveal (but only about 2% more ASs). These findings point out the need for an increased awareness of and a more critical attitude toward the applicability and completeness of given data sets at hand when establishing the generality of any particular observations about the Internet.