Is it possible for me to wake up from anesthesia before i'm supposed to?

Very Unlikely: It would be very unusual for you to have any memories from during a surgery, however, it is possible. Anesthesiologists use your vital signs and a variety of monitors to help ensure this does not happen. They are well trained to prevent such things from occurring.

Unlikely: It is theoretically possible but highly unlikely. You will have a member of the anesthesia department monitoring just you and doing nothing else for the entire operation. That's what they are there for, to keep you safe and comfortable.

Yes: It's possible, but you shouldn't, in the hands of a well-trained professional. If your anesthesia provider is vigilant, they can easily tell that you're waking up long before you might be aware of it, and take steps to change that.