English is an analytical language. That means that the grammatical nature of a word that you see will be determined by the role it plays in the sentence and the sentence's syntax. The same word could be a verb or a noun, or an adjunct noun. In the case of an analytical language, you cannot determine the role of a word without seeing the whole sentence.
Specifically, for your examples, if the preceding noun of the two is not an adjunct noun, it is an object of the subordinate clause, and thus another verb is coming, because the main clause has to have a predicate.
The path students saw... was rough and bumpy.
(The path which students saw (or "The path seen by the students") was such-and such.)
If the preceding noun of the two is an adjunct noun, then it's all one noun phrase which is the subject of the sentence, and "was" is the predicate of that sentence, so no other verb is forthcoming.
That said, even if one sees a sentence (or part of the sentence) without a context, one tries to imagine which text it would fit into nicely. In the case of "The path students saw...", if we were talking about some students that have seen a specific path (after all, there is a "the" in front of the path), then it would be in a text that is telling a story about those students, and the students would have been already described and would have required their own "the".
The path the students saw was rough and bumpy.
(As part of a continuation of a story about a group of students who, say, have gone on a hike in a forest.)
The absence of "the" in front of "students" does make me (in this particular example) think that this sentence is about "the students of the path" (e.g., I don't know, students that have come to some guru to study his "Path").
But the main way to determine these things is to have a complete sentence and assign S-V-O status to its various components.