Well, right now, we're in a place where The Internet really only facilitates greater efficiency in telecommunications, and does so in such a way that can be distributed and encapsulated across many independent partitions.
In other words, whatever we would be capable of accomplishing with pen and ink on paper, and carrier pigeons, or smoke signals; that's what the internet does, but at nearly the speed of light, for volumes of data beyond anything worth attempting as a physical implementation.
In that sense, the only thing that denial of service really accomplishes for a hermit state like North Korea (which presumably attempts to censor the external internet for it's non-elite commoners already), is such that they lose face on "the world stage" where they receive no respect anyway.
The boundaries where things start to get ugly, in a new and truly modern sense, would be circumstances where autonomous weapons platforms run rampant, and inflict wide-spread death and destruction in various theaters of conflict at scales of their own choosing. I don't think that's a reality yet, at least not without a nuclear exchange. Drones, for the most part, are still essentially remote-controlled vehicles, operated by humans, particularly with respect to the decision to use force.
A scaled back version of that, which we might see emerge, before autonomous robots are used to crush a nation like it were a load of dirty laundry (or rather, before robots start to decide for themselves, which nations, or regions to crush), is infrastructure attacks that cripple things like electric and water services for extended periods, triggering cascades of famine and disease. For that to occur, a country would have to foolishly place all of its eggs in one basket, and lay prone to catastrophic failure without proper redundancies in place.