Ideal power level to suit a driver?

[Pbassred]

Do drivers have an ideal power level? One that they sound best at?

I changed my 10" no-name chinese drivers out of my little PA cabs and installed Eminence Betas instead (Greater handleing power). These cabs aren't the most musical in the world. (I quite often see them outside shops where they sound awful.) I use them as part of a larger system so its never my intention to use them below 500Hz or above 3000KHz. All the same, I expected them to sound a little more "full". If anything they sound more mid, and less bassy, but this was only on a lounge size amplifier at discreate volume. I'm wondering if they would sound different if they were driven a little harder.