TITLE: A $\mathbb{R}^{n}$ -fiber bundle which do not admit a n-dimensional vector bundle structure
QUESTION [5 upvotes]: Is there  a  fiber  bundle $(E,B, \mathbb{R}^{n})$, with typical  fiber $\mathbb{R}^{n}$,  such that there  is  no any  $n$-dimensional vector  bundle  structure on the  pair  $(E,B)$? That is  there is  no  a continuous  map  $p:E \to B$  such that the triple $(E,B,P)$ would  be  a $n$ dimensional vector  bundle.

REPLY [6 votes]: Yes, there a lots of them. 
The topological tangent bundle $\tau$ of any topological non-smoothable manifold $M$ gives you an example since a vector bundle reduction of $\tau$ is by smoothing theory equivalent to a smoothing of $M$.