Let $\cdots \to X_2\to X_1$ be a Postnikov tower for an $n$-dimensional CW-complex $X$. Given some $k<n$, is it possible to find a CW-complex $Y$, such that $\dim(Y)\leq k$ and with a Postnikov tower in which $Y_k\simeq X_k$?
If the case $k=n-1$ is somehow special, then I'd like to know more about that case.
The inclusion $X_{k+1}\rightarrow X$ of the $(k+1)$-skeleton of $X$ is $(k+1)$-connected. In particular this inclusion induces isomorphisms on $\pi_l$ for $l\leq k$, so that $P_l(X_{k+1})\simeq P_l(X)$ for $l\leq k$. If $X$ has no $(k+1)$-cells then $X_{k+1}=X_k$ is a CW complex of dimension $k$ for which $P_k(X_k)\simeq P_k(X)$, so if we take $Y=X_{k+1}=X_k$ then we are done.
More generally, if all $(k+1)$-cells are attached along homotopically trivial maps $S^k\rightarrow X_k$, then $X_{k+1}=X_k\vee \bigvee S^{k+1}$ and the induced map $\pi_kX_k\rightarrow \pi_kX$ is an isomorphism. Hence in this case you can take $Y=X_k$ and be done.
In fact under the assumption that $k>1$ these are the only possibilities that can occur for which such a $k$-dimensional complex $Y$ as in the question can exist.
For assume that $X$ is a complex of dimension $n$ and $Y$ is complex of dimension $k<n$ such that both the Postnikov sections $P_k(X)$ and $P_k(Y)$ are homotopy equivalent. In this case we can assume without loss of generality that $P_l(X)=P_l(Y)$ for $l\leq k$.
Then in particular $X\rightarrow P_k(X)=P_k(Y)$ is $(k+1)$-connected, and since $Y$ is of dimension $k$ there exists a map $f:Y\rightarrow X$ lifting $Y\rightarrow P_k(Y)$. By construction this map induces isomorphisms on $\pi_l$ for $l\leq k$. Therefore there exists a CW structure on $X$ in which $X$ is obtained from $Y$ by attaching cells of dimension $>k$. That is $Y$ is a $k$-skeleton of $X$.
If this CW structure does not require $(k+1)$-cells then $X_{k+1}= X_k=Y$ and we are done.
On the other hand, if this CW structures does require $(k+1)$-cells then there is a cofibration sequence $$\bigvee S^k\rightarrow Y=X_k\rightarrow X_{k+1}$$ obtained by choosing attaching maps for these cells. If $k>1$ then there is a Blaker-Massey exact sequence starting $$\pi_k(\bigvee S^k)\rightarrow\pi_kY\rightarrow\pi_kX_{k+1}\rightarrow 0$$ and extending to the right. Since $\pi_kX_{k+1}\cong\pi_kX$ by induction, and $\pi_kY\cong\pi_kX$, the left-hand map in the above sequence (which is induced by the attaching map of the $(k+1)$-cells) must be the zero homomorphism.
It follows from this that $X$ admits a CW structure with $$X_{k+1}\simeq Y\vee \bigvee S^{k+1}.$$