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.