Is "He is very rich and he is not happy" correct?

Why

Karan is correct that "and" can grammatically join almost any two clauses, and the sentence isn't technically broken — but conjunctions do more than just glue two ideas together, they also signal HOW those ideas relate to each other. "And" tells the reader the two ideas simply add on to each other, going the same direction: "he is rich and he is generous" feels natural because both ideas point the same way. "But" signals a contrast, a turn against expectation: being rich usually makes people expect happiness, so pairing rich with "not happy" is a genuine surprise, exactly the kind of turn "but" exists to flag. Using "and" here quietly hides that surprise, making the sentence read flatter than it should. Test: read both versions aloud — "rich AND not happy" reads like a neutral list, while "rich BUT not happy" reads like it's pointing at something worth noticing. His sister is right: "but" signals the contrast between wealth and unhappiness far more clearly than "and" does, even though "and" isn't strictly ungrammatical.