No, that wouldn't usually happen. I've had surgery a few times. What they do is monitor your brain activity and pulse rate to see if your are coming out of anesthesia for some reason. If you were suffering, at the very least your pulse would go up, and they will notice.
Indeed there is several monitors that can be used to measure “awareness” including entropy, bi-spectral index, and SEDline to name a few.
SEDline is my personal favourite as you can see when a patient has complete “burst suppressions” - this is a total absence of brain activity at all and has been associated with post anesthetic delirium particularly in elderly patients.
