The Qur'an itself does not discuss the topic of possession, in the sense of a non-human being taking over a human being, apart from emphasizing that the prophet is not insane/possessed (majnun). That it, it seems, at the time, people would receive or claim to receive messages from jinn or other beings, perhaps sometimes while possessed. The Qur'an does not argue that this is impossible; rather, it argues that the Prophet is above this, and that the revelation is directly from God.

Beyond that, one can glean from the fact that the Qur'an mentions humans and jinn as two halves of creation (thaqalayn) that there is some reason for us to know about each other and some sort of interaction, even though it is usually not visible. 

In practice, possession - in the sense of ceding one's self to an external being - is extremely rare, except in religious traditions where people put forth considerable efforts to induce this condition. (Of course, one can debate whether this is actually happening or not.) 

For most people, the most that might happen is that they might be somewhat influenced or affected by a being external to them (such as the waswas of Shaytan). If this does happen, it is not necessarily a sign of moral weakness or sinning; there are all sorts of factors that could contribute to it. (However, violence and licentiousness may also be contributing factors) Similarly, salat and du'a helps to avoid or mitigate this, although it is not an absolute guarantee that it will not happen. 

In any case, it is better not to be paranoid about these things unless one actually has a solid reason to think they have a problem. How often do jinn get blamed for the acts of men!