Let $X,Y,Z$ be connected manifolds and consider the fiber bundles $$\pi_1: X\rightarrow Y\\ \pi_2: Y\rightarrow Z\\ \pi_3 : X\rightarrow Z$$ such that $\pi_3=\pi _2\circ \pi_1$. Now if the fiber of $\pi_3$ is connected. Can the fiber of $\pi_2$ be discrete?
I think it is not possible since, say $\pi_2^{-1}\{z\}=\{y_1,y_2\}$ then $\pi_1^{-1}\{y_1,y_2\}$ is equal to two distinct fibers in $X$. Unless $\pi_1$ is injective?
If $\pi_2^{-1}(z)$ is discrete, then $\pi_1:\pi_3^{-1}(z)\rightarrow \pi_2^{-1}(z)$ is a surjective map onto a discrete set, and so $\pi_3^{-1}(z)$ is disconnected.