It depends on the details.
For each styrofoam board the bowling ball falls through, two things happen: the ball is slowed by the board, and the ball is accelerated by gravity. In energy terms, you can say the ball expends (loses) kinetic energy by straining and breaking the styrofoam, and gains kinetic energy from the conversion of potential energy as it falls.
Once you know what happens for the first board, you can predict the rest. But you need to observe more than just whether or not the first board breaks. You need to consider the speed the ball is falling. Especially, whether it is moving slower or faster the moment it meets the second board, than the moment it met the first board.
If the ball has slowed down, then it was more slowed by the board than it was sped up by gravity (that is, if it loses more kinetic energy to the board than it gains from the potential energy converted to kinetic energy in falling that distance), and it could eventually slow to a stop. Whether or not it does so before it gets all the way to the ground depends on how much it's slowed, how fast it was going in the first place, and how many boards there are. if it does reach the ground, then making the stack sufficiently higher would prevent it from doing so.
If the ball is going faster after the first board (that is, it gains more kinetic energy from converted potential energy than it expended on the board), then it will not stop no matter how high the stack of boards is. In that case making the structure taller by adding more boards could not change the outcome.
Changing the spacing of the boards could affect the outcome, though, because that changes the ratio of energy required to break a board versus potential energy converted by falling the distance of one board-to-board space. Changing the thickness or strength of the boards would also change that ratio.
Respectfully,
Myriad