That's an accurate description of the ARPANet (from which the Internet was derived), but not the Internet.
The ARPANet wasn't designed for commercial or consumer service, competing backbone providers, the present size of the Internet, etc. etc. etc.
That aside, the DNS system has always been the weak point failure-wise, and I assume that's what they are talking about.
Wikipedia gives a good short summary of how close we have come in the past to total failure: