Ah… I think I am misremembering. It appears Dennet discussed the “possibility” but only in order to discuss the ways we could know that this is in fact not what’s happening.
-FrL-
ETA Reading further, it looks like he’s making some more complicated point, but for this thread, the basic takeaway is that, no, he’s not claiming that what the OP proposed might in fact be true. He does seem to be saying something like it might be sort of true, though. 
Not only can you be completely awake and feel all the excruciating pain but you also can remember it. Many people have had their entire life destroyed by the horror of this experience.
http://boards.straightdope.com/sdmb/showthread.php?t=434109&highlight=anesthesia
And, as has been mentioned, there are ways: Restraints, even really professionally-applied ones, leave marks, and fighting someone makes you short of breath in a way you aren’t when you awaken from a general. Also, it would be difficult to impossible to fully soundproof every clinic and OR so nobody could hear the screaming. Finally, it would have to be a massive global conspiracy requiring every MD, APRN (CRNAs are only one subset), RN, LPN, tech, and the teachers thereof to be actively complicit and utterly sociopathic. Conspiracies are fragile things that tend to get broken for the stupidest reasons. Massive global conspiracies are quadratically more fragile (n[sup]2[/sup] connections for n nodes, given that n is large indeed) and involve many times more stupid people. Add in the fact the conspiracy would have to have been running uninterrupted for decades and yet there is still enough of a flaw to allow us to talk about it here and the implicitly proposed scenario is too stupid to take seriously.
Derleth, you have to think people on this thread are really stupid to post what you just did.
See post number 20 please.
-FrL-
There are dissociative agents which dissociate pain signals from the part of the brain that cares about it, so to speak. These drugs tend to leave the patient more aware and are useful because they tend to leave intact important reflexes such as the protection of the airway. When we use these drugs, it’s common for patients to not remember much of a particular procedure even though they were speaking and breathing while the procedure was being done. In my practice in the Emergency Department we use these drugs commonly for such minor procedures as fracture-dislocation reductions. The amnesia is useful and sometimes surprising–a patient who seemed to be momentarily in pain will return to normal sensorium smiling and completely amnestic for the event once the drug wears off. (The term “excruciating agony” is not, of course, in our vocabulary 
even if there was some light temporary screaming.) It is of medical value to us to use these agents because we can do painful procedures without discomfort to the patient and without putting them into such a deep state of unconsciousness that they require more complicated and dangerous maneuvers such as intubation or general anesthesia (GA).
The anti-pain effect of commonly-used amnestic agents such as Versed (midazolam) is different from that of narcotic analgesics such as Dilaudid (hydromorphone). Versed doesn’t really seem to deaden pain; it just makes you not care, and perhaps not remember. Dilaudid will obtund you to where you don’t care, but when you arouse the patient enough, they will still complain that they are in pain. Neither is an anesthetic in the strict sense of the term–neither one actually blocks the pain signals to the brain. (Both drugs in excess will render you reasonably close to dead, so we’re talking about clinical doses here…)
Modern general anesthesia uses a combination of drugs to produce mental relaxation (call this “not caring” if you like), unconsciousness and paralysis, and the amnesia is really a result of deep hypnosis preventing consiousness from even creating a memory rather than a direct amnestic effect.
Most drugs that produce very deep unconsciousness will prevent memory from being laid down, so that the patient is unaware of anything and therefore unable to create memory.
It is possible for paralytics to allow a patient to remain aware but be incapable of physically reacting (although sometimes various autonomic responses signal that the patient is mentally awake but paralyzed…) and the extreme versions of these cases are the horror stories regarding GA gone awry. I am unaware of the use of drugs to produce a retrograde amnesia to cover this up (per Der Trih’s comment). Certainly if a patient were found to be aware during GA, some of the drugs used immediately to correct the situation would include those with an amnestic effect, however.
Many drugs produce a state of mental relaxation and many of them render memory for the event either unreliable or non-existent. We titrate most of the drugs we use for either partial (conscious) sedation or full general anesthesia because individual responses vary so much.
The whole GA cycle is sort of a continuing cocktail: Give the patient something to help them relax, then some drugs to prevent certain physiologic responses, then a rapid induction agent with paralytics and short-acting hypnotics to facilitate intubation, then agents used for longer-term deep hypnosis (full GA).
First, there’s no need for restraints because they administer paralytic drugs ( like curare ) along with the drugs to render you unconscious. Even unconscious, they don’t want to risk a reflex twitch while the surgeon is working on you.
Second, the OP isn’t claiming any “conspiracy”, but is asking if a claim he/she heard is true.