Fiber bundles with same total spaces, but different base spaces (may be silly question but), Does there exist two fiber bundles (or in particular vector bundles) whose total spaces are the same but base spaces are different?

Sure, consider the space $\mathbb{S}^1 \times \mathbb{R}$ with the projections $\pi_1: \mathbb{S}^1 \times \mathbb{R} \to \mathbb{S}^1$ (a trivial fiber bundle with fiber $\mathbb{R}$) and $\pi_2: \mathbb{S}^1 \times \mathbb{R} \to \mathbb{R}$ (a trivial fiber bundle with fiber $\mathbb{S}^1$). For an example of (trivial) vector bundles, take the total space to be, say,  $\mathbb{R}^3$, written as a product space $\mathbb{R}^2 \times \mathbb{R}$. Again, the natural projections give two vector bundles over different base spaces.   
