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*][1], 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][2].


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).    








[1]:http://books.google.fr/books?id=h-wc3TnZMCcC&printsec=frontcover&dq=spanier+algebraic+topology&hl=fr&ei=_15lTp61C8Pj4QTU0vDHCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCsQ6AEwAA#v=onepage&q&f=false

[2]:http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Jerzak.pdf