Necessary yes. Sufficient no.
Necessary
John 6:44 beings with "οὐδεὶς δύναται" No one has this ability
The Greek verb is δύναμαι *("dunamai"), meaning: to be able to, to be possible, to have the power (source). It's a pretty clear way to say "this is not possible".
The same verb (dunamai) defines a necessary condition in John 3:5:
Except a man be born of water and of the Spirit, he cannot enter
into the kingdom of God.
--
Sufficient
In the previous section we identified at least 3 necessary conditions:
- Be drawn by the Father
- Be born of water
- Be born of the spirit
If there are at least 2 necessary conditions, then no single necessary condition is sufficient on its own (we could delve into other arguments for additional necessary conditions, but we've already met the de minimis threshold of at least 2).
Furthermore, multiple passages show that it is possible to fall, even after being drawn by God, even after being baptized, etc.
From John 17:12 we see this applied to Judas:
those that thou gavest me I have kept, and none of them is lost, but
the son of perdition
Dottard has already noted the poignant passage in Hebrews 6:4-6:
4 For it is impossible for those who were once enlightened, and have
tasted of the heavenly gift, and were made partakers of the Holy
Ghost,
5 And have tasted the good word of God, and the powers of the world to
come,
6 If they shall fall away, to renew them again unto repentance; seeing
they crucify to themselves the Son of God afresh, and put him to an
open shame.
Paul (or Luke, or Barnabas, or Priscilla, whoever wrote Hebrews, take your pick) gives a stark warning to those who, like Judas, came to Jesus, knew His words, received His ordinances, and then turn against Him.
--
Conclusion
John speaks of coming as a result of drawing, but coming alone was not sufficient. Coming and staying is what counts.
Addendum
What happens with someone who is never drawn...Is there any chance of salvation...?
This is an extensive topic in itself, but I suggest that the short answer is:
- There are those who never learn of Jesus Christ in this life
- There are none who never learn of Jesus Christ in this life OR in the realm of spirits
IE
- ~ This life = may be TRUE, but
- (~ This life ^ ~ In the spiritual realm) = FALSE
I discuss this in more detail, drawing from 1 Peter 3:18-20, 4:6, and how these passages were understood by the early Patristics, in this post, making the case that no one will eternally be unreached; everyone will have that opportunity.
(this does not mean all will get more than 1 opportunity, nor does it mean all will accept)