That’s different than just asking your sergeant and being let go. You’re referencing a failure-to-adapt chapter separation which requires medical, behavioral health, chaplain, and career counseling over several weeks. It’s possible, and I’ve only seen a handful over 19+years. The Army doesn’t do it often because everyone thinks they’ve made a mistake about a week in to basic training. They don’t want it to happen often because then everyone would request it.
In my USAF basic in 1981, one of the guys cried all night, the first night. He asked to go home in the morning. He was 18, and the very first night he spent without his parents in the house.
Rather than give him a discharge, they gave him orders to go home, and report back in three months.