Bouncing is a non-scientific term we use here to describe the phenomenon of a kitty overreacting to situations such as:
1. BG dropping too low during the cycle
2. BG falling to lower levels than kitty's system is accustomed to even if those BGs aren't that low;
3. a more rapid than usual drop in BG even if the BGs are still fairly high.
The kitty's body interprets these situations as a BG emergency and through a complex set of processes involving the liver, etc. converts glycogen ("stored glucose") into glucose and dumps it into the bloodstream to raise BG in order to compensate. The problem is that it's often an overcompensation - a bounce. BGs will go higher than expected and if the too high insulin dose continues, BGs will stay stuck up there. That's what fools you into thinking the dose is too low.
If high BG is the result of bouncing then a lowered dose will cause lowering of BG. That's why we use the protocol here of testing before AM and PM shots as well as at least one other time in a cycle and also why we recommend starting at a low dose (usually 1 unit twice a day) and raising it slowly by 0.25 unit at a time so you don't jump right up past the good dose. Vets often prescribe too high a dose to start - eg. 2 units twice a day - and you might already be well past kitty's good dose.
As far as numbers stabilizing goes, that takes time. Achieving regulation where a dose keeps BGs in a good range most of the day every day takes time. There's a lot of trial and error involved and, as we always remind people, it's a marathon, not a sprint. Patience is essential.