There's no such thing as a hard deadline. Everything is made up. What your manager is saying is: "To be successful in the market, I think I need to have X thing in Y days."
It's the engineer's responsibility to provide information and estimates that will shape the business's direction. If you don't clarify or negotiate requirements based on your technical knowledge and experience and just take requirements as law written in stone you are not providing the true value of an engineer.
If you are doing that - pushing back and grounding requirements in reality - and the business plows ahead with unrealistic goals anyway, then that simply means the people you work with are unreasonable and you should seek employment elsewhere. If they're not even willing to listen to people they're paying to have expertise then they're probably reading the market wrong too and I wouldn't trust them to be around long anyway.