SMD ferrite beads - DC current rating
Two phenomena will affect things here. 

Firstly, magnetic effects. An increase in current will cause an increased flux density inside the ferrite core. At a high enough current - likely around the nominal 3A rating, this increase in flux density will approach a point where the material's relative permeability will start to decrease - saturation, which you mentioned in your question. This will result in a lower inductance, and hence a lower AC impedance. There is also the phenomenon of "field popping" described in this SE answer - but it is not directly relevant to your question.

Secondly, thermal effects. It is unlikely that the ferrite will fuse at 3A - although it will at some current higher than this. Before that happens, however, the ferrite will heat up, causing an increase in DC resistance. This will also affect the permeability of the ferrite core, and hence the AC impedance of the part. This effect isn't trivial to predict, but it is actually likely to increase the AC impedance (over a range from room temperature up to high tens of degrees C). Take a look at this Google image search for some typical permeability vs temperature charts for ferrite materials.

So we have three different effects - a decrease in AC impedance, and increase in DC resistance and an increase in AC impedance. So what dominates? This will depend on the exact makeup of the inductor, but in general you will see an overall decrease in AC impedance. This is a bad thing (in general).