Humans are bastards but I can not see people just falling into total anarchy for decades to come when something bad happens. When bad happens looting occurs, but it's a temporary solution for people who bear a grudge to their neighbor or want a TV. Looting and Raiding has never been a way of life, save for the occasional highwayman appearing in story and legend.
My point is that it just can't go on for centuries upon centuries.
It all depends on just how bad the war was, really. If it was truly as all-encompassing as depicted in the FO universe, then yes, the world probably would have relatively quickly fallen into utter barbarism and ignorance, forgetting and losing most of what had been gained since the dark ages. Most of our modern civilization is based upon just a few key things which we all now take for granted- easy access to food, water, electricity, and fuel. All else is just fluff, when it comes down to it. You destroy the national power grid, fuel supply and hence food delivery system (not to mention the food itself, due to nuclear winter and fallout poisoning all the natural foods), and the survivors from the bombs pretty quickly starve out. Canned and dry foodstocks in stores and warehouses which survived the bombs and radiation, would pretty quickly get used up during the millions of survivor's desperate (and doomed) fight against starvation.
Large parts of the country (world) which are pretty dry and get most of their drinking water delivered from pumping deep wells or by long-distance irrigation systems dependent on lots of electricity and continuous good maintenance by people hundreds of miles away- pfft. They're flocked. Would the competition for water and foodstocks be organized and peacefull, when you have millions of survivors dying of hunger and armed with the hundreds of millions of weapons legally owned by Americans? Heh. It wouldn't be pretty. If you had a hoarde of canned goods in your basemant, and the armed, starving mob wandering down the street found out about it, what do you think is going to happen? Mhmm.
People just don't understand the awful power of fear and hunger and thirst, if it suddenly became a universal, immediate, life-and-death imperative for everyone on earth- with only your brain and your own two hands to do anything about it. And when most of those millions have died off, and the nuclear winter finally eased up years later, and the lucky and resourceful handfuls of survivors crawled out of their fortified basemants and caves to go about the job of surviving and repopulating the world, they aren't going to have huge piles of food still laying around in all the (remaining) warehouses and stores of the world, with which to get by on. The mass starvation in the months immediately after the war took care of most of those. No, they are going to have to hunt whatever animals are left, and learn to plow and plant whatever they can, just to stay alive.
With nothing but hard, back-breaking work keeping them alive from then on, and no medical infrastructure and few supplies left, and few doctors, life expectancies will plummet. Surviving 40 years will be a long life, and many diseases, accidents and misfortunes will just be fatal or permanently debilitating. Child mortality will be awful, especially considering the huge amount of radiation the world and all its inhabitants will have absorbed by that point. Within several generations, there will be nobody left who remembers life before The War. How much of the education, knowledge, and technology will the 'old timers' have been able to permanently impart onto their descendants, during those harsh times when every hand old enough to lift a hoe will be needed in the fields just to keep the community/family alive? Bullets become pretty undependable after 15 to 20 years of sitting around, and there would come a time when any still laying around simply won't work anymore. Weapons that don't require a lot of technology to create and use would have to be reverted to, for hunting and defense.
Within 100 years, or probably much less, mankind could very easily devolve again into a primitive way of life that europeans from the midieval darkages of our history would have been right at home in.
Some people just assume that much of our current way of life would just mysteriously continue on as if nothing happened, after such a worldwide holocaust. They are wrong. Dead wrong. Take away all the basic underpinnings for civilization, and civilization will cease to exist, pretty quickly.