Suppose $Z\to X,\ Z\to Y$ are covering spaces.
Does there exist a space $W$ s.t. $X\to W$ and $Y\to W$ are covering spaces?
Reference:
Exercise 1.3.11 in page 80 from Allen Hatcher's book Algebraic topology gives an answer.
Exercise 1.3.11
Construct finite graphs $X_1$ and $X_2$ having a common finite-sheeted covering space $\widetilde X_1 = \widetilde X_2$, but such that there is no space having both $X_1$ and $X_2$ as covering spaces.
Best Answer
Let $X$ and $Y$ be the two graphs having two vertices and three edges. There is a common two-sheeted covering space $Z$ of $X$ and $Y$ which is a graph with four vertices and six edges. Exercise: Find $Z$ and show that $X$ and $Y$ are not covering spaces of any other spaces (besides themselves, of course).
Interesting side note: The covering spaces $Z\to X$ and $Z\to Y$ are defined by free actions of ${\mathbb Z}_2$ on $Z$. These generate a ${\mathbb Z}_2\times{\mathbb Z}_2$ action, but it is not a free action since the product of the two generators has a fixed point.