This is something that has been indeed an interesting issue that has been raised in recent years. The answer is an unequivocal yes.
Firstly, Jesus or Yeshua himself, came from Jewish stock. He was born a Jewish woman, raised in a Jewish family, worshipped the Father like a Jew, kept the feasts a Jew etc. This needs to be considered.
Next, The disciples themselves were also Jews, from different walks of life, such as tax collection, fishing and other jobs in the ancient Israel.
Another point to note, Throughout the NT. Putting aside their inspiration for a brief moment but looking at them purely as historical documents, we observe in Paul's letters he addressed both Jew and Gentile congregations, mixed congregations I might add.
In particular, in his letters of Romans and Colossians, he mentions the subject of not judging believers on peripheral issues such as Sabbath, Food Laws, New Moons etc. The subject of this can be found here in this article and I speak in depth on certain issues raised by Messianics and even some groups such as 7th Day Adventists:
Also, the unknown writer of Hebrews wrote to his fellow Jewish believers to encourage them to continue in the faith in Yeshua and not to turn away, lest they fall into destruction.
Some heretics such as the Ebionites even adhered to Jewish customs, but of course they denied Paul and had taught various other heresies. Keith Thompson has made a note of their heresies and quotes various scholars to tell us what they believed. Read section 2 of his article to get a better look into them: http://answering-islam.org/authors/thompson/paul-historical.html
Putting aside that in modern day Messianic congregations you have Gentiles who have converted to Messianic Judaism and putting aside how much orthodoxy a Rabbinic Jew held to before conversion to Messianic Judaism, lets focus on the fact that there are Jews today that believe in Christ and Jews in the days of the apostles who did.
Those Jews do not lose their identity as Jews and remain Jews despite their faith in Christ, because he himself is Jewish, that doesn't change. Though is Jewish, this doesn't mean Gentiles are spiritual Jews, that idea is ridiculous. If one is a Jew or Gentile when they become a Christian, they remain such. A Gentile however does have to turn from idols and paganism nonetheless.
I am aware that later in church history the church pressed for either Jews renounce the practices of the TANAKH or remain as Jews without the Messiah, but there is no evidence in the NT that you cannot be a Jew and follow Jesus, despite the protests of Rabbinic Jews and Christendom, the NT makes it clear there are JEW and GENTILE congregations that mingle together in the Messiah.
Think on these issues.