I should explain a bit more about the Domestic Reserve List. It is the limit of minor leaguers you can have rostered: 165 in the regular season. This started up a few years ago and was actually reduced from 180 to 165 last year, so last year is when things started getting pretty tight. I've got the tigers around 162 right now (but haven't given it a scrub in a while). Players in the majors don't count; players in the minors on the 40 man do count. Players on the minor league 60-Day IL and full season IL do not count, nor do suspended or restricted list players. You can only have 15 players on the minor league 60-Day IL at one time, and the Full Season IL is only for certain surgeries and prohibits a player from certain activities if they are placed on it.
Last year, a bulk of the cuts around the draft were low level players or mid level third catchers. There's lots of players many on here haven't heard of who are just taking up spots on rosters who would be released well before the Kreidlers and Workmans of the world are.
Also, injuries have a way of making roster crunches work out. There's ways to juggle playing time beyond that.