The short answer, IMO, is "it depends".
The higher the level the competition, the tougher it is. I know of a couple of guys that have earned a roster spot at the University of Tennessee in recent years based on their performance in fall tryouts. However they are the exception rather than the rule.
DIIIs by definition, are "walkon". There are no athletic scholarships associated with a DIII. You might find some DIIs and NAIA programs that are as tough a D1s. Others might be easier. One problem with smaller, private, schools is that they often have much higher tuitions. Therefore you potentially have a larger financial burden to deal with.
However, if you can demonstrate that you are good enough, then you can earn a spot.
It would also depend on what position you play and how that school's depth chart stacks up. I'd certainly want to know how many scholarship players there are at your position.
Many of the walkons at D1 are "invited" or "preferred". These players are often considered "good enough" to be scholarship players. But, with the scholarship limitations in place for baseball, it's not uncommon for players to pay their own way if they really want to play for a particular program.
While you should have some realistic expectations about your abilities, and the situation at the school, I'd always encourage you to chase your dream. Just be reasonable about it. If there are particular programs you are interested in, I'd suggest contacting the coaching staff and inquire about their policies. You might also post a question on this board about a particular school. There's a good chance someone on this board can give you some information.
Good luck.