How many charts?

Question

My question is rather vaque but I hope that You will feel what I would like to know. A manifold (say: real, $n$-dimensional) is something which locally looks like an open set in $\mathbb{R}^n$. A rank $k$-vector bundle over $M$ is a bundle of $k$-dimensional linear spaces which is 'locally trivial'i.e. looks (locally) like $U \times \mathbb{R}^k$. You can cover a manifold with charts domains and you can cover it with open sets over which a given bundle becomes trivial. My question is the following: are there any relations between the smallest number of (both) possible coverings? More interesting would be the behaviour of these numbers when varying vector bundles: are these numbers bounded when:
- the rank of the bundles is fixed and bundles are varying,
- the rank of the bundles could also vary?
Any other properties and results similar to this problem would be interesting for me.