Is the composition of two bundle projections necessarily a bundle projection? That is, if $f: X \rightarrow Y$ and $g:Y \rightarrow Z$ are bundle projections, is $g \circ f: X \rightarrow Z$ a bundle projection? Assume $X$, $Y$ and $Z$ are manifolds.
Here is what I know. The answer is affirmative when (1) $f$ is a covering map and $g$ is bundle projection; (2) $f$ is a bundle projection and $g$ is a covering map of finite degree.
What can we say about the most general situation? 
Thanks.
 A: In the category of finite dimensional manifolds, I proved this in a very long paper (p. 23): the composition of smooth fiber bundle maps is a smooth fiber bundle map. Spanier's counterexample is not a finite dimensional manifold.
A: I think that the following works: Let $X\to Y$ and $Y\to Z$ be locally trivial
fibrations with all spaces paracompact and $Z$ locally contractible (I do not assume that a
fibration implies that all fibres are homeo- or diffeomorphic). We want to show
that $X\to Z$ is locally trivial. We may then reduce to the case when $Z$ is
contractible and $Y=Z\times F$. Under the paracompactness assumption locally trivial
fibrations are homotopy invariant (see for instance Husemoller: Fibre bundles
GTM 20, Springer, Thm 9.8 plus a simple reduction to the principal bundle
case).  Hence there is a locally trivial fibre bundle $X'\to F$ such that $X\to Y$ is
isomorphic to $Z\times X'$ which is the local trivialtity.
A: No, the composition of two bundle projections needn't be a bundle projection.
 It is already not true that the composition of two covering maps is a covering map.
You can find a counterexample in Spanier's classic Algebraic Topology, Chapter 2, §2, Example 8, page 69. 
Another counterexample, based (!) on the notorious Hawaiian ring space is given by Jerzac's very  nice,detailed paper here.
However, on the positive side, the composition   $g\circ f :X\to Z$   of the covering maps $f:X\to Y$ and $g:Y\to Z$
is a covering map in each of the following two cases:
a) The covering $g$ is finite (= has finite fibers)
b)  The space  $Z$ has a universal covering  (for connected $Z$, this means that  $Z$ is  locally pathwise connected and semi-locally simply connected). For example, CW-complexes  have a universal covering space, since they are even locally contractible.   
