Yes; the sentence given:
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"The students were bored. They didn't know this history would be important to them."
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could be correct.
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The word: "history" has different meanings, including: "a particular systemic account of events". As such, the term: "this history" be referring to such a particular account or reference. (And note: the word "history" has a plural: "histories"). So depending on the context, the sentence could be entirely correct.