Does a cofiber sequence of CW complexes induce a cofiber sequence of skeleta? Suppose $X\to Y \to Z$ is a cofiber sequence of CW complexes. We can replace the maps with homotopic cellular maps $X_n\to Y_n \to Z_n$ taking the $n$-skeleton of $X$ to the $n$-skeleton of $Y$ and similarly for the second map. 
Is the resulting sequence $X_n\to Y_n\to Z_n$ necessarily a cofiber sequence? I feel the answer must be no, but a counterexample is eluding me. 

No. For example, $S^1 \to \bullet \to S^2$ is a cofiber sequence, but it doesn't remain a cofiber sequence after taking $1$-skeleta. 
