This scenario of "nuclear apocalypse by inadvertence" would require some inordinate incompetence at some point. Namely, we can imagine a buggy router which mixes some packets together, and sends the wrong packet to the wrong destination. And then, inexplicably, the military system which receives the packet which, by a stroke of bad luck, contains what that system interprets as the coded order for "it's payback time, commies !" just goes ahead with it.
Normally (hopefully), command systems which can trigger the nuclear Ragnarok are supposed to resist wanton attacks from bad people with evil intentions. They use authentication and cryptography and dual control and many other things to prevent outsiders from launching missiles, even if they try real hard to craft packets which will mimic the normal launch packets. The scenario envisioned here is that a random bit flip in a router due to a cosmic ray just succeeded in mutating or redirecting an innocent game-related chunk of data into a one-way-ticket for fission inferno. In other words, the scenario assumes that all of the following hold true:
- the security systems and failsafes on the controls for nuclear missiles are so awfully weak that a single misrouted packet is sufficient to trigger the launch;
- none of the existing attackers, be they spies from other nations, nihilists who crave emptiness, or bored students who should know better, succeeded in doing a task so simple that a random bug achieved it (it's like being crushed at StarCraft, when your opponent is a cat who decided to take a nap on the keyboard; it is very embarrassing);
- the nuclear strike force controls are connected to the Internet.
Any of these conditions requires a lot of people to be terrible at their own job, to a magnitude which is remarkable even if we consider that some of them are a public organization, and others are pre-pubescent geeks. The whole picture is then quite improbable.
It may happen, but a lot of improbable things are more plausible than that, and arguably worse, including the usual example of a ten-kilometer asteroid striking the Earth, something which has already happened in the past, will probably happen in the future, and totally dwarfs out our puny nuclear arsenals when it comes to destructive consequences. Such an event occurs on average about once every 40 millions of years, i.e. has probability 1 in 40 millions to occur every year. Anything which could happen with a lower probability is therefore negligible.
A generic name for such theories could be "misaligned danger assessment". It is worrying about the very improbable instead of the very probable. If you want to live longer, don't worry about nuclear missiles; instead, quite smoking, drink less alcohol, and eat more vegetables and less fat.