I did Bleak Falls Barrow very early on with my character and didn't have a problem, but I ran into this problem on other doors similar to the bleak falls one with the claws and the 3 rings. I think there are quite a few claw door puzzles through out the game, as I probably have 6 or 7 of those claws at this point. I don't think the glitch is specific to bleak falls barrow, but can just happen on the 3 ring doors randomly. the one I had it on I ended up finally getting it by leaving the cave and re-entering a couple times and then if finally started to work. on another one, I think I actually was trying to go through a door too early in a quest. it was the Galdur amulet one, and I went to the final cave just randomly when I walked by it and it would not let me in, but once I finished the other parts of that quest and I got back to that cave it let me in...
All that said, those puzzle doors do seem to be glitchy from time to time. possible fixes are leave and come back, or try loading a previous save and going in again. I've made that my standard procedure when buying houses is to save right before buying the house and upgrades, and then testing all storage, weapons racks, and mannequins before going to far. if anything is glitchy, I reload the save prior to buying the house and upgrades and do it all again until everything works... once I know they work I don't seem to have any problems moving forward...