I had this problem on my second playthrough. I had over 2000 xp and couldn't level up even if there were no enemies nearby. I found a suggestion on wikia to delete all of my game saves (including) the auto save and complete the tutorial first thing on my new playthrough. As soon as I finished the tutorial, it was suggested on wikia to immediately go to the outskirts of Goodsprings (before you get 200 xp), where the game asks you if you want to revise your character. I did this and was able to level up like normal. I play on 360. I apologize if my explanation is confusing.
Actually, your answer is insightful and not confusing at all.
I never realized that it might be possible to get out into the game world without passing the Character Recreation Check. That may be what is happening to the OP. It certainly makes sense.
I'm not sure what deleting the saves has to do with it. Just go back and make sure to save after getting the Check.