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 as soon as $Z$ has a universal covering map (that is as soon as $Z$ is connected, locally pathwise connected and semi-locally simply connected).