Short version: Why can't cryptanalysis methods be carried over to science, which looks like a trivial problem by comparison, since nature doesn't intelligently remove patterns from our observations? Or are these methods already carried over?
Long version: Okay, I was going to spell this all out with a lot of text, but it started ballooning, so I'm just going to put it in chart form.
Here is what I see as the mapping from cryptography to science (or epistemology in general). I want to know what goes in the "???" spot, and why it hasn't been used for any natural phenomenon less complex than the most complex broken cipher. (Sorry, couldn't figure out how to center it.)
EDIT: Removed "(cipher known)" requirement on 2nd- and 3rd-to-last rows because the scientific analog can be searching for either natural laws or constants.
Oh. Well that renders most of my response irrelevant. Then the answer is "probably yes". Getting the basics of the grammar won't take much effort. So one can tentatively identify which words are verbs, nouns, adjectives, etc. Assuming that you aren't going out of your way to be terse with your speech, there will be a fair bit of redundancy that should in a large enough text become obvious. For example, if we've identified how one says "and" in the language or at least the version for nouns, then might be able to identify the plural form for verbs, or something close to that. Moreover, if we see a word that frequently shows up near the word for "and" we could tentatively guess that that was a word for two as a cardinal number. Similarly, one might be able to get three as a cardinal number. This gets a handle on your number system.
In the direct context of Navajo which you used as your other example, one also has a correspondence with physical events which if one had that data could potentially help a lot.
So if a time-traveling mischief maker gave the NSA a copy of "The Klingon Hamlet" in 1980 (minus all the English text), would they have been able to "decrypt" it?