Conversational implicature.

However, CP is often violated.

 Since CP is regulative, CP can be violated.

 Violation of CP and its maxims leads to conversational implicature.

In real life, our frequent disregard of the maxim is generally less blatant and the implicatures more subtle. When people say more than seems warranted, we might suspect them of 'beating about the bush' so as to impress us, or hide something from us under their verbiage. When people are sparing with their commendation, we suspect them of 'damning with faint praise'.

Regulating how much language we use to conform to the quantity maxim, and exploiting the maxim to create implicatures to add further significance to what we say, are complex and elusive processes.