I have a friend who does this - she and her husband lost some weight, and certainly getting off sugars and processed foods is an excellent thing.
But on the negative side, they are just sick all the time, catching everything. (To be fair, they had some problems beforehand too - lots of colds, flu, etc.). SO that tells me that they just aren't getting the right cellular nutrition. Th problem is that they just can't go anywhere - our group has a pot luck meal a few times a year, and it becomes a huge problem of identifying what's acceptable. Part of this is because there are different definitions of what "paleo" is.
The other problem I see, as a nutritional consultant and educator, is that the meats they are eating are just not what was eaten in paleolithic times. Ancient meat was not corn-fed animals loaded with antibiotics and raised in CAFOs (Confined Animal Feeding Operation). Even our definitions of "free range" or "cage free" are not consistent or regulated. The foods (fruits, veggies etc.) we eat today are nutrient deficient even if they are organic (and the "policing" of organic foods is spotty).
I agree there are problems with our grains as well - the wheat of today is not those "amber waves" from 50 years ago! I also know that the vast majority of people should be able to eat all foods in moderation - this huge increase in food sensitivities and "allergies" is unnatural, avoidable and correctable. I've worked with dozens and dozens of people who've been avoiding certain foods, and in most cases they can go back to eating them. Those with an anaphylactic reaction may not be able to eat that food, but with the right techniques, they can at least be near them (can be in the house of someone who eats them, can eat out in a restaurant, etc.) or not have a life-threatening reaction if they are accidentally exposed. Once people understand that an autoimmune response to a normally harmless food (soy, wheat, egg, nuts, you name it) is unnecessary and can be repaired/eliminated, the need for these elimination diets goes away.