When one first encounters the concept of vector field, especially in physics, it is often presented just as n-tuple of numbers $(x_1, x_2, \ldots , x_n)$ prescribed to each point. In this manner $n$ is allowed to accept arbitrary value.
However, when one proceeds with vector fields in differential geometry, the dimension of a manifold dictates the dimension of a tangent space:
$$\dim M = \dim T_x(M)$$
There are other ways to attach a vector space to each point of manifolds --- tensor products of tangent spaces, p-forms, etc.
Now I ask, given an n-dimensional manifold, what are allowable dimensions for vector spaces on it? Are all of them allowed, or there is no way to naturally construct a k-dimensional vector field (in a broad sense) for certain k?
Another (stronger) version for this question, given n-dimenstional vector space, can we naturally construct k-dimensional vector space from it for arbitrary k?