Self-awareness determines how consciously "in tune" you are with your current reality. Lucid dreamers aim to be highly self-aware while awake and while dreaming, for the greatest frequency of lucid dreams. But the modern world is so distracting, most people are not very self-aware at all.
And this comes into play in the paradox of a false awakening. The dream of awakening is highly vivid, suggesting a high level of self-awareness (for a dream, at least). Yet many false awakenings go unrecognized; assumed to be waking reality, there is absolutely no awareness that it's all a dream.
A false awakening may involve getting up, having breakfast, getting dressed, heading out for work... all the things you do every day on autopilot. It can all appear all too real and solid to warrant questioning its authenticity. Indeed, you will only appreciate how real a false awakening is when it happens to you - it reveals the remarkable capacity of the human brain to emulate reality.