False. When you wake up in the wee hours of the morning for no apparent reason (no loud noises, the house is not on fire, etc) there is only a 7% chance that you have woken up because someone/something is staring at you.
However, there is a 50% chance that you woke up because some kind of bug crawled on you and made you itch or sneeze, a 23% chance you were awakened by a small earthquake, a tremor, if you live in eathquake country, a 68% chance that you woke due to a dream you were having (you may or may not remember the dream) and a 92% chance you woke up because you're really thirsty and/or you desperately need to pee.
That "I feel like someone is watching me" feeling is very seldom due to the fact that someone is actually watching you. It is, however, an indication that something in your immediate environment ain't right and you would do well to sit up and take notice whenever you realize you feel this way, especially if the feeling comes on in a sudden and overpowering way. If you were a cave man feeling this way it would be because your cave was about to collapse on you and kill you or perhaps due to a flash flood or a forest fire intent on doing the same. In this day and age it's usually because you're about to get hit by a bus, there is a tornado approaching your home or perhaps the batteries in your carbon monoxide detector have died and you need to get up and open some damn windows right effin now. It's even possible (though highly unlikely) that someone is actually watching you. Either way, when you get that "I'm being watched" feeling you need to be very aware of exactly what's going on around you because there is an 85% chance you might need to take cover, evacuate the building or otherwise tuck & roll your asss out of harm's way.
Your intuition is NOT to be ignored, it is there to help you survive, avoid harm, eat well, mate often and to otherwise help you meet your needs while simultaneously helping you to avoid pain, injury and death. Those who heed their intuition (gut feelings) often live better and longer than those who do not.