Can an Apostille Be Correct but the Foreign Authority Still Reject the Document?

Yes, and it's one of the situations that confuses people the most when they get a rejection: the apostille is perfectly valid, the authority that issued it had jurisdiction over the document, and the foreign authority still rejects it. We already flagged it as one of the ten problems in our checklist on why was my apostille rejected?; here we explain why it happens and what to do about it.
The apostille certifies that the signature or seal on the document is authentic. It does not certify or guarantee that the content, format, age, or any other substantive requirement of the document meets what the receiving authority requires. A rejection under these circumstances is a document problem, not an apostille problem.