It depends on a lot of things. Once you are drafted, the guys who loved you and drafted you have no more say on things. The development side of the organization takes over and if someone on that side is not as high on you it can cause difficulties for a player. I am aware of an All-American player who was drafted in the 5th round and received a 200k signing bonus. He has played two seasons now of short-season ball and that cannot be a good sign for him in that organization. He also has name players ahead of him like Tony Pena's son so a lot can be defined by who you have to leap over.
For lower round picks, production will set you free. It appeared to me that they go about two or so weeks where everyone gets a chance. For the prospects, they seem to continue to get decent playing opportunities even when they struggle. For non-prospects, you might find yourself sitting on the bench for long periods of time if failing to produce.
The worst thing a player can do is leave college early and find they are behind someone in some organization or worse yet, sitting on the bench. Unless ridiculous money is involved, it can happen to just about anyone. Injuries obviously play a big part in all this. If you stay in college four years, unless you are a very high draft choice, the money is almost nill. It is hard to make a perfect decision sometimes.