Yes, and it is usually less fixed than it sounds, particularly if no safe plan exists yet. Ask the case manager directly what happens if no placement is confirmed by the target date, and involve the hospital's patient advocate if the timeline feels unsafe.
Inside the discharge window
Hospitals are under real financial and regulatory pressure to move patients toward discharge, but that pressure does not override a hospital's own obligation not to discharge someone into an unsafe situation. If a placement genuinely is not ready, say so plainly and ask what the hospital's contingency plan is -- most have one, even if it isn't offered up front. Every hospital has a patient advocate or ombudsman role of its own, separate from the long-term-care ombudsman, and they exist specifically to intervene when a family feels a discharge is being rushed.