It is known - although I personally don't know the proof - that if $X$ is a finite dimensional smooth real manifold of dimension $n$, then there is a smooth finite atlas of $X$ with $n+1$ charts, although they might be disconnected (actually, I know this is true for topological manifolds; is this still true for smooth structures?).
Now let $\pi:Y\rightarrow X$ be a smooth fibered manifold (i.e. $\pi$ is a surjective submersion) with $\dim X=n$ and $\dim Y=n+m$. Then $Y$ has a smooth atlas consisting of fibered charts, i.e. charts of the form $(U,x^1,\dots,x^n,y^1,\dots,y^m)$, where the $x^i$ ($i=1,\dots,n$) are constant when restricted to each fibre.
My question is, does $Y$ also have a finite atlas consisting of fibered charts?