A worldbuilding thought:
Consider a region where one particular language is the first language of a vast majority of the inhabitants. Now say something happens (anything from political events to social stratification to some cataclysmic geographical change) and two sets of speakers of this language are mostly, but not completely, isolated from one another. (continued)
Let's say that for one reason or another, the lexicon of this language is very fixed, but the grammatical structure is not. Perhaps they have a very well-standardized orthography but very few style guides and grammar books. The influence of the small-but-nonzero amount of contact with the other group would help to keep them similar as well.
Now let the two populations evolve their language over a long period of time, a few centuries or so. (continued)
Over time, the grammatical structure of the languages may change. Consider the possibility that both populations' grammar diverges in very different directions. Now, you're left with two distinct languages, both with all the same words and the same (or mostly the same) meanings of those words, but the syntax, sentence structure, and grammatical rules are very different.
What would such a pair of languages be like? How intelligible would one be to a native speaker of the other?
A few centuries probably isn't enough time for a language's grammar to change this drastically, so let's just pretend I said millennia instead of centuries there.