Let $X$ be CW complex having only cells up to dimension $n$.
I want to prove that every $m$-dimensional vector bundle $E$ over $X$ decomposes as a sum $E\cong A\oplus B$ where $A$ is a $n$-dimensional (perhaps $n+1$?) vector bundle and $B$ is a trivial bundle $X\times \mathbb{R}^{m-n}$.
I remember vaguely that I've seen such a statement somewhere but I have no idea how to prove it. My first thought was to use the cellular approximation theorem an show that every map $X\to Gr(m,\infty)$ is homotopic to a map with image in $Gr(n,\infty)$ but I have doubts now that the cell structure on $Gr(n,\infty)$ can be choosen in such a way.