Code talkers.

[Rankin, Robert L]

I'd guess that, over time, the "code(s)" could have been broken.  As several have said, any language can be reduced to its grammar.  In the case of the American Indian languages, they were used as tactical codes, not strategic codes, as far as I know.  The latter are used for diplomatic and general staff, etc. communications, and it was those that our cryptanalysts had the best luck with against the Japanese and Germans.  Even if the Japanese had painfully learned Navajo (or Seminole, Choctaw, Omaha, Comanche, etc.), they would have had to be able to understand the language spoken rapidly by native speakers under battlefield conditions.  How many of us could do that with the Siouan languages we've studied for so many years!?  As an aside, I don't think the alleged "difficulties" of the Navajo language had much of anything to do with its success as a code.
 
On top of the language there was a fairly simple substitution code with different words for various military terms, and they would have had to be decrypted too.  All in all, it was a terrifically efficient system, and the participants deserve all of the credit that has been bestowed on them, however belatedly.  
 
The contribution of the Navajos has become pretty well known, but I think someone should try to interview any left who were from other tribes -- in any war.  And, if not them, then their living relatives.  There's definitely a book to be written there, and who better to write it than someone with roots in the Native American community or a linguist?
 
Bob