The Internet ? No. The Arpanet ? Also no. But SAGE was - its prototype (the Cape Cod air defense project) was a packet-switching network and SAGE itself was as well... And staying up on doomsday for a bit of nuclear combat was the essence of SAGE's functional specification. SAGE pioneered most of the concepts and technologies of the Arpanet, whose purpose was absolutely not combat, so it is easy to imagine how the nuclear strike resilient Internet urban legend evolved.