What you're describing sounds very much like sleep paralysis. It happens to many people, and is frequently perceived as an attack of sorts. Basically, a person's brain wakes up just enough to be mildly aware of their surroundings, but it is still in a sleep state, and the rest of the body is not woken up. There is also frequent experience of hypnogogic state during the same time. That is, the brain is still in a somewhat dream-like state. It can result in hallucinations.
Compound everything, and the inability to move voluntary muscle groups paired with hallucinations from the hypnogogic state, result in perceptions such as a weight on the chest or limbs, and a presence or even a figure in the room.