It's not.. Basically your body part is made numb.. and on top of that your body is administered with anaesthesia so you will feel groggy and be in a blur state. That one's called 'twilight anaesthesia'. You can read more here
http://www.wisegeek.com/what-is-twilight-anesthesia.htm
I can probably say that in almost all cases when the person says he/she feels pain halfway through the surgery.. it's not that the LA wore off, it's just the twilight part that did. When it wears out, the patient becomes more aware of what's going on. That's all.. people are just freaked out wondering why they're suddenly conscious of what's happening and.. attribute that to the so-called "LA" wearing off. But I don't deny that in some cases the LA does really wear off.. but that's quite rare.
Do you really think that if your nose's getting cut up and the LA wore off, you can still lie on the bed? You'ld be screaming your lungs out in pain man
And for most minor cosmetic surgeries, doctors do not usually use GA. The post recovery period after a GA op is always much worse than LA, cos of the residual anaesthesia gas still inside the body.. and the patient won't feel good for many hours after the surgery. Plus GA comes with risks.. so pls don't think that GA is always better than LA.