Was just coming in to report this as well. Same thing happened with a character last night. Got to level 16, went to select a perk, clicked back to check something on the Skills screen, then bam: unable to select a Perk in the menu (Done is greyed out). I tried a bunch of other things including earlier saves, then much earlier saves. Thought it be something weird with having obtained the Boss (level 30) achievement the night before.
Once I ruled out a few things, I figured that either the Boss achievement does something wonky to the level cap or that my character (not my save, because I tried several for that character and they were all broken at the Perk menu) had become corrupted. I threw in the towel on that character, started a new one and then went to bed. However the same thing just occurred with the new character at level 4 (there goes my level cap hypothesis).
The good news is that I think I know what caused it. I selected a perk, then clicked Back to check my skills, then Continue and noticed the same thing. I'm almost certain that clicking Back once you're on the Perk menu will corrupt the character (again, I just tested by rolling back to a level 3 save and advancing the level 4 again).
To summarize: it appears that clicking Back from the Perk menu corrupts the character file and makes it impossible progress the game.
I'll see if I can test further. Hopefully the folks that were perusing these forums before the previous two patches are still doing so.
EDIT: Ok, went into the XBOX memory and removed all the save files. Recreated a new character with the same name as the aforementioned level 4 character. Cleared level 2 okay but had the same problem again at level 4. Exited the game, cleared the cache (LB+RB+X, right?), and tried again. Same thing at level 4.
Going to repeat the steps above with a different character name and see what happens. My guess is that the character file is stored via name somewhere that isn't visible (like the save files) and that the game is registering a new character with an "existing" name as a continuation.
Obviously feedback from someone at Bethesda would helpful while I beat my head against the wall trying to figure this out. I'll throw in root cause anolysis charts if it would help