Paul writes (Romans 8:38-39, NIV)
For I am convinced that neither death nor life, neither angels nor demons, neither the present nor the future, nor any powers, neither height nor depth, nor anything else in all creation, will be able to separate us from the love of God that is in Christ Jesus our Lord.
and again (Ephesians 6:11-12)
Put on the full armor of God, so that you can take your stand against the devil’s schemes. For our struggle is not against flesh and blood, but against the rulers, against the authorities, against the powers of this dark world and against the spiritual forces of evil in the heavenly realms.
All people, including Christians, are continually subjected to temptation (demonic assault) but Christians are uniquely able to resist because of the Spirit dwelling within them. This does not mean that Christians always will resist - otherwise Paul would have no need to continually remind people to refrain from sin - but only that they have a special capacity to do so.
Augustine says (City of God, 10.22)
For the devil cannot conquer or subdue any but those who are in league with sin, and therefore he is conquered in the name of Him who assumed humanity [...]
But also, discussing the Apocalypse and the idea that the devil will be chained and then loosed, writes (20.7, 20.8 respectively)
Let no one be dismayed by the circumstance that the devil often seduces even those who have been regenerated in Christ, and begun to walk in God's way. For "the Lord knoweth them that are His," and of these the devil seduces none to eternal damnation.
The Almighty does not absolutely seclude the saints from his temptation, but shelters only their inner man, where faith resides, that by outward temptation they may grow in grace.
None of this extends to "possession", if regarded as the total subjugation of the human will to demonic influence. That would mean that the possessed person no longer had the capacity to turn towards Christ, whereas Christian belief is that anyone may be saved, even at the moment of death. Augustine's view is that the devil being in you does not mean that he has control, but rather that you are suffering a kind of spiritual contamination which creates a tendency to sin. You are the devil's captive but Christ can always release you.
Dante famously included in the Divine Comedy a region of Hell where souls may go even before the death of the body (Inferno 33:124-132, Longfellow translation):
Such an advantage has this Ptolomaea,
That oftentimes the soul descendeth here
Sooner than Atropos in motion sets it.
And, that thou mayest more willingly remove
From off my countenance these glassy tears,
Know that as soon as any soul betrays
As I have done, his body by a demon
Is taken from him, who thereafter rules it,
Until his time has wholly been revolved.
The commentaries I have read on the Comedy regard this as an innovation of Dante's own. That would suggest that complete demonic control of the living was not previously believed to be possible.