While, for the most part, you can recruit most students into Golden Deers, you won't be able to recruit everyone, people such as Dimitri and Dedue from the Blue Lions or Edelgard and Hubert from Black Eagles as these characters are incredibly important to their respective routes story. Besides them, there are a few exceptions as well as in order to recruit students, you have to get them to a certain bond level. Some of these bond levels cant be achieved until after major story events, well after the time skip, such as Casper, and Gilbert, as at that point, it'll already be too late. This issue can be avoided in new game plus as you can buy bond levels for students there.