Sorry, no, they actually didn't.
First, let's look at divinely revealed revelation to the disciples:
Matthew 16:13-17
King James Version
13 When Jesus came into the coasts of Caesarea Philippi, he asked his disciples, saying, Whom do men say that I the Son of man am?
14 And they said, Some say that thou art John the Baptist: some, Elias; and others, Jeremias, or one of the prophets.
15 He saith unto them, But whom say ye that I am?
16 And Simon Peter answered and said, Thou art the Christ, the Son of the living God.
17 And Jesus answered and said unto him, Blessed art thou, Simon Barjona: for flesh and blood hath not revealed it unto thee, but my Father which is in heaven.
The Father revealed to the disciples that Jesus was the Christ, the Son of the Living God.
Born again?
Let's go a little further:
Matthew 16:20-23
King James Version
20 Then charged he his disciples that they should tell no man that he was Jesus the Christ.
First, ask yourself why the Lord charges His disciples not to tell men what had been revealed to them?
Then ask yourself why it is at this late time in His Ministry that the Lord begins to give a clear Gospel presentation to His disciples:
21 From that time forth began Jesus to shew unto his disciples, how that he must go unto Jerusalem, and suffer many things of the elders and chief priests and scribes, and be killed, and be raised again the third day.
22 Then Peter took him, and began to rebuke him, saying, Be it far from thee, Lord: this shall not be unto thee.
23 But he turned, and said unto Peter, Get thee behind me, Satan: thou art an offence unto me: for thou savourest not the things that be of God, but those that be of men.
A moment of weakness for Peter? No, it's just a matter that the Gospel of Christ is still a Mystery. It is not being revealed.
Truly God the Father revealed to them Jesus was in fact the long-awaited Messiah and that He was the Son of God, but He did not reveal to them that Christ would die for their sin so that they might receive eternal life.
This is a record of Peter denying the Gospel of Jesus Christ.
Unless you want to say that men can be in Christ and have eternal life and reject the Gospel then you are going to have to change your perspective on the disciples (and Old Testament Saints) being regenerate.
Next, let's take a look at another example of the unbelief of the disciples:
John 16:28-32
King James Version
28 I came forth from the Father, and am come into the world: again, I leave the world, and go to the Father.
29 His disciples said unto him, Lo, now speakest thou plainly, and speakest no proverb.
30 Now are we sure that thou knowest all things, and needest not that any man should ask thee: by this we believe that thou camest forth from God.
31 Jesus answered them, Do ye now believe?
32 Behold, the hour cometh, yea, is now come, that ye shall be scattered, every man to his own, and shall leave me alone: and yet I am not alone, because the Father is with me.
If Jesus can question the belief of the disciples I see no problem doing it myself. Especially when being born again is dependent on belief in Jesus Christ Himself and the Comforter is not yet revealing the Gospel Mystery.
Did they have faith in Jesus? Of course, but they did not understand the Gospel.
The Old Testament Saints were justified based on their response to the revelation they received, but they all died not receiving the promises (plural).