Every year in the mid summer, young man from Mostar (people call them "Liske" - leafs) have organized jumping off the Old Bridge. As the time passed by, skill of fight against highness and swift, cold Neretva has been improved. Only several places in the world can be proud of such spectacle.
Although the ruins of the Old Bridge are still under reconstruction, jumps have been lately performed from the platform built on the remains of the Bridge.
There are some "Leafs" who will jump off for you if you pay them 10 (in summer time) or 20 (in winter time) Euros.
Neretva River is very fast mountain river. It is ideal for water sport as canoeing or kayaking. There is Canoeing Club close to Green market in Old City (left bank). Maybe you can go there and ask to rent a canoe.