Is there any embedding theorem for fibre bundles?

Question

I would like to know whether there is an embedding theorem for fibre bundles, like Whitney embedding theorem. When can a given fibre bundle be a subbundle of some higher dimensional bundle?

Answer 1

Yes, there are analogues of Whitney's embedding theorem for vector bundles.
For example: if $X$ is a compact topological space, any real vector bundle on $X$ is a direct summand (and thus a fortiori a sub-bundle) of a trivial vector bundle $X\times \mathbb R^N$.
This was proved by Swan in 1962 in this article, Lemma 5, page 268.

There are many similar results obtained by relaxing the condition that $X$ be compact or by passing from the topological category to the $C^k$ or holomorphic or algebraic or...categories.
Actually the first theorem of this sort was proved by Serre for algebraic vector bundles on affine varieties in his immortal article Faisceaux Algébriques Cohérents, always affectionately nicknamed FAC .

Edit
As an answer to some comments below, let me add the following:
If $X$ is paracompact of finite covering dimension $d$ (for example a manifold of dimension $d$) then every vector bundle of constant rank $r$ is indeed a summand of a trivial bundle of finite rank $N$.
This is stated in Milnor-Stasheff, page 71.
I guess that one can't bound the rank $N$ by a function of $r$ and $d$ alone : $N$ might be arbitrarily large for very twisted bundles.