The code was almost certainly unbreakable when it was invented and used back in the 1940s, but I am sure given the current super-computers and all the power the NSA has, that the code could be broken in a matter of minutes. If this was not the case, they would still be using such systems now.
Yes, someone can eventually break Chester Nez's code, as great as it is, because it is created by humans, and you have to be able to figure it out at some level, otherwise it wouldn't be helpful from a military standpoint. He did a great job, though. His code will be hard to break.
Now that the world knows the basis of Chester Nez's code, that it was based on the Navajo spoken language and used Navajo words for animals etc to represent common language used in warfare, it should now be possible for someone who has knowledge of the Navajo language to break the code.
The reason the Navajo language was chosen in the first place is because it was nearly impossible to learn if you haven't been raise hearing it. There was also no written form making it all the more impossible to crack. I think it is one of those great pieces of history that will never fully be understood.