When you're blackout drunk, or under anesthesia, your brain can still infer connections between sequential stimuli (e.g. you can understand words), but none of those memories stay in your brain long-term. I'd say that we were actually conscious during that period, but in the present moment we have no recollection because our memory (for some chemical reason related to alcohol/anesthesia) wasn't stored properly for long-term use.
I imagine the same would happen if we somehow managed to "remove" memories from our minds. After the removal, our current self would not have any recollection of those memories removed - as if we weren't conscious during that period whatsoever.
In conclusion: the feeling of consciousness is deeply related to continuous storage of our life-long memories.