I think it may be good to first create a definition for the general addition of tangent vectors at different points, which may help illustrate when/how/if this is possible. Without this definition it seems too speculative to be able to answer more definitive questions. EDIT: it seems like the first "natural" definition would be that of choosing a vector space isomorphism to identify any two spaces. But then there goes the naturality issue. This would bring you to the concept of a connection.Sorry, I meant to put fibre, not section (was just about to change it, but you beat me).
Would it be correct to say that one could add a vector from a tangent space at one point to a vector from a tangent space at another point, but this will in general not correspond to tangent vector in either space - it will not be tangent to any curves passing through either point?
Also, from another point of view could one argue that adding the two vectors component wise relies on choosing a basis and is this clearly coordinate dependent, hence such an operation has no geometrical meaning - the resulting vector will not be in the tangent bundle over the manifold?!
Sorry to labour the point, I don't know why I'm finding it so hard to conceptualise.