Usually, yes, but you have to read the Bible from the point of view of the audience to which it was written.
1. Everyone would agree that being born of water and the Spirit (John 3:5) is the same thing as being “born again” (John 3:3). So what does it mean to be born again? We see what it means in Romans 6:4: “Therefore we have been buried with Him through baptism into death, so that as Christ was raised from the dead through the glory of the Father, so we too might walk in newness of life.” The newness of life is a reference to being born again.
No, not everyone would agree that being born of water and the Spirit (Jn. 3;5) refers to the new birth. However, let us take it from the perspective of the audience it was spoken and written unto. Nicodemus was a "ruler" of the Jews or part of the Sanhedrin. Water was repeatedly used in the Temple for various ceremonial cleansings (cleansing of the lepers, etc.). The Priest's used the laver repititously. Water was a common theme for ceremonial cleansing in nearly all the temple rituals. The temple and all of its ordinances were established after a heavenly pattern. The temple with all of its ordinances were TYPES rather than the antitype. What was water in its use of ceremonial ordinances designed to be a type of?
We know that sacrifices were the original divine external ordinances used by God since the foundation of the world (Heb. 11:4). We know the language of redemption commonly characterized these sacrifices as they were offered "for sin" and "for cleansing" but the New Testament commentary on Old Testaments sacrifices deny that they ever could remove sin literally (Heb. 10:4) but were only "shadows" of the antitype (Heb. 10:1) that does remove sin literally (Heb. 10:5-18) once and for all.
Therefore, no Jew could deny that "water" was an external material element whereas the "Spirit" was not an external material element. What "water" was used as a type on the external part of the temple, its furniture, priests, cereminal defilements, the Spirit was the antitype that literally regenerated, cleansed through the Word of God the internal spiritual aspect of man.
The sacrifices were external types that ceremonial obtained remission of sins, but that was merely in "shadows" or type, as it is the Lord Jesus on the cross that literally obtained remission of sins.
If anyone could see the connection between water and the Spirit or sacrifices and the Messiah it should be a "ruler" of the Jew that was well versed in the Levitical ceremonial system practiced in the Temple.
However, we also realize the current Jewish cultural system in the day of Christ had perverted the truth of the gospel and turned the type into the antitype for obtaining justification and salvation (Acts 15). The Catholic church continues this perversion by the Jewish cultural traditions of confusing the type with the antitype.
In the very next chapter (Jn 4) Jesus clearly makes use of the typical design God had behind the use of "water" when he told the Samaritan woman that the "water" she was drawing from the well was symbolic of a spiritual well springing up with in her unto eternal life, and yet there was no immersion into the well demanded to obtain that reality. In John 13 where Jesus again uses water as an analogy of spiritual cleansing with the ceremonial backdrop of the Priests who had to bath before putting on their priestly garments and then after that merely wash their feet and hands, clearly an analogy to the new birth by the Spirit and then daily cleansing (hands, feet) by confessing their sins. The cleansing of the lepers in Luke 5:12-15 is another clear analogy that ceremonial cleansing by the preists did not literally remove any literal defilement as they were already literally cleansed from the defilement of leprosy by the spoken word of Christ. Rather, the ceremonial cleansing by the Priests through the ceremonial use of water was "for a testimony" of literal salvation from leprosy.
"Saved by Water" in the case of the ark is a "like" type of baptism, in that both were pictures of the resurrection of Jesus Christ, which provides literal salvation (1 Pet. 3:18-21).
What Catholicism has done is taken the outward symbol and made it part of the literal antitype, thus repudiating the truth of the gospel.
John the Baptist insisted upon "fruit" of repentance before administering baptism (Mt. 3:6-8) demonstrating that the outward act of baptism did not effect or coincide with any inward grace or salvation but was conditioned upon repentance and faith in the gospel of Christ, and John did preach the gospel of Christ (Jn. 3:36; Acts 19:4).
So to be told that one must be born "of water and the Spirit" in the mind of Jew who was an expert in the Temple ordinances and use of water would make perfect sense in an analogous manner of "born of type and antitype" or "born of water EVEN the Spirit." This analogous use is clear in John 4 with the use of water coming from the well. However, Catholicism and the historical antecedents of the developmental stages of Catholicism (Ante-Nicene Fathers) committed the fundemental error of confusing the type with the antitype (Acts 15)and produced a false type of Christianity equal to the false type of Judaism in the time of Christ.