Both Bromine and Iodine are Halogens, group VII on the Periodic Table. Generally, the higher you go on this column, the more reactive the halogen is, and more reactive elements tend to replace less reactive elements in ionic compounds.
In elements that form positive Ions (cations), the lower they are on the table, the more reactive they are, as their valence (outer) electrons are further from the nuclei, and thus less strongly attracted. They lose these electrons more easily resulting in more vigorous reactions, particularly displacement reactions.
In elements that form negative Ions (anions), the higher they are on the table the more reactive they are, as their valence (outer) electrons are closer to the nucleus, thus gained electrons are pulled in more readily. They gain electrons more easily resulting in more vigorous reactions, particularly displacement reactions.
If you put Sodium Iodide in Bromine (a more reactive halogen than Iodine), the Bromine should react with the Iodine forming Sodium Bromide crystals and Iodine gas/solid depending on the ambient temperature of the Bromine container. You should see a color change, and observe a temperature change due to the reaction. The reaction should be relatively slow due to the stability of Sodium and the closeness of Iodine and Bromine on the Periodic table.