So I've changed my mind! I think the answer is "yes". Suppose $X$ and $N$ are given. Let $\epsilon>0$ and let $M$ be such that $\mu({x:N(x)>M})<\epsilon$. Now build a Rokhlin tower with height $M/\epsilon$ and error set of size at most $\epsilon$. Let $A$ denote the base of the tower. For each $x\in A$, let $n_1(x)$ be the least integer (up to $M/\epsilon-M$) such that $N(x)\le M$. Then let $n_2(x)$ be the least integer greater than $n_1(x)+N(T^{n_1(x)}x)$ (up to $M/\epsilon-M$) such that $N(T^{n_2(x)})\le M$ etc. For each $x\in A$, there are $k(x)\ge 0$ $n_j(x)$ defined. 

Let $B_x=\{T^{n_j(x)}:j\le k(x)\}$ and let $B=\bigcup_{x\in A} B_x$. This is a measurable set. I claim it has the properties that you want.