Let’s talk about baseball, since that seems to be the sport in question. Most D3 teams are going to settle in at around 30. If the positions were evenly disbursed, which they are not, you’d have room for 3 to play each position. Obviously they are not evenly divided, and you are likely to find 20 field players and 10 pitchers (give or take).
If you assume in the NESCACs, the teams are given two slots and three tips (again, this can vary) there is room for a walk-on or two each year. There also is room for a cut or two. Then there is attrition and injury . . … If a stud walks on, they will find him a spot. But there is absolutely no guarantee that if you try out, you will make the team. Remember, the level of play at most of the academic D3s (Cal Tech excluded) is a significant jump up from high school.
The take-away is that, yes, teams can and often do have room beyond the tips and slots. Your question is will every baseball team have extra room, and the answer to that is “no.” It really just depends. You need to talk to the coach to determine what the needs for any given year are. For example, the Claremont teams seem to over-recruit. That is because their coaches don’t have that much sway with admissions. The same is true for Chicago and MIT. So, if if your kid has the academic chops to get in at one of those on his own (no mean feat), POSSIBLY there is a better chance to walk-on that at other schools. By contrast, I would think walking onto one of the Little Three would be a tough go, as they tend to perform well on a consistent basis, and plenty of good players really want to play for those teams.
It can happen, however. A kid from our area did not do the baseball showcase circuit and did not play on an uber competitive club team. That said, he was extraordinarily fast. He made a D3 baseball team and pinch ran for a year, before quitting to focus on track. But he was very fast, and for that to work, you would need quite a bit below a 7.0 60.
If a baseball team has had a down year, and a down recruiting year, it is POSSIBLE that an admissions office might suggest that you speak to the coach. If that does not happen, I believe the adcom will look at the baseball experience in the same way it would any other EC, like the school play or band. A leadership position on a high school team, such as captain, probably would be received like any other high school leadership position.