First, let's formulate the question properly: 

Given a topological space $X$, define $d(X)$ to be the supremum of numbers $n$ such that $X$ is homeomorphic to $Y\times {\mathbb R}^n$. 

Then, working in the topological category, you are asking if there is a manifold $X$ and a number $k$ such that $d(X\times {\mathbb R}^k)> d(X)+ k$. 

(The question is equivalent to yours since every principal ${\mathbb R^n}$-bundle with the total space $X$ is the same thing as a proper   ${\mathbb R^n}$-action on $X$; at the same time, each principal ${\mathbb R^n}$-bundle is trivial.) 

Now, here is an example: Let $X$ be the Whitehead manifold (or any other contractible 3-manifold not homeomorphic to ${\mathbb R}^3$). Then $d(X)=0$. On the other hand, $X\times {\mathbb R}$ is homeomorphic to ${\mathbb R}^4$ (see for instance [here](https://mathoverflow.net/questions/424165/how-to-prove-the-product-of-whitehead-manifold-and-mathbbr-is-homeomorphic)), hence, $d(X\times {\mathbb R})=4$. 

The same example works in the smooth category.