Here is the definition of vector bundle of bounded geometry from Shubin: Spectral theory of elliptic operators on non-compact manifolds, page 29:
Let $E$ be a complex vector bundle on a manifold $X$. we shall say that $E$ is a bundle of bounded geometry if it is supplied by an additional structure: trivializations of $E$ on every canonical coordinate neighbourhood $E$ such that the corresponding matrix transition functions $g_{UU'}$ on all intersections $E \cap U'$ of such neighbourhoods are $C^\infty$-bounded, i.e. all their derivatives $\partial ^\alpha_y g_{UU'}(y)$ with respect to the canonical coordinates are bounded with bounds $C_\alpha$ which do not depend on the chosen pair $U$, $U'$.
The same definition with different wording can be found in Eldering: Normally Hyperbolic Invariant Manifolds: The Noncompact Case.
Can anyone give an example of a vector bundle that is not of bounded geometry? I have trouble thinking of one, as rescaling a trivialization of my bundle gives another trivialization smaller norm transition functions.