R. Cohen proved the immersion conjecture in a 1985 Annals paper:

Cohen, Ralph L., The immersion conjecture for differentiable manifolds, Ann. Math. (2) 122, 237-328 (1985). ZBL0592.57022.

Any smooth compact n-dimensional manifold admits an immersion into Euclidean space of dimension 2n-a(n), where a(n) is the number of 1's in the binary decomposition of n.

Is there any result of this kind for vector bundles E over compact manifolds? Notice that the sphere bundle of E is compact. Maybe there is a silly argument...