Connectivity Problem
Maximize Edge Disjoint Paths (EDP)

The problem is NP Hard and here are some results:

1) It accepts $O(\sqrt{m})$ approximation by Kleinberg in 96 in general graphs.

2) The above is tight for directed graphs (Guruswami et. al.).

3) Inapproximation result by Andrews, Zhang : O($log ^{\frac{1}{2}-\epsilon}n$).

4) For planar graphs, Chekuri et. al. has found a O(1) approx with any edge being in O(1) paths. 

I will update with references later but you can google out.

EDIT: There is a recent paper by Chekuri in 06 which has improvised it to $O(\sqrt(n))$ in undirected graphs. You will get the above references in that paper. The paper is at 

theoryofcomputing.org/articles/v002a007/v002a007.ps.gz