So, does the Bible bear out this teaching? My answer, for what it's worth, is yes. Let me give you an annotated list of scriptural citations which support the thought:
1) Revelation 7: note that the 144,000 are described in earthly terms, whereas the Gentile saints are described in heavenly ones;Perhaps no chapter of scripture is more helpful eschatologically than Revelation 12. Once one properly understands the symbols, the end-time scenario clarifies and the timing of end-time events settles into place. The imagery of the woman clearly hearkens back to Joseph's dream, the figure is obviously Jewish. That she was pregnant brings into focus two thoughts: 1) she would give birth to something like her, and 2) while in the womb that something was expected but hidden. If ones sees the woman as a corporate symbol (like the nation of Israel), rather than an individual (like Mary), then the infant must taken the same way.
2) Matthew 24:32-51: note that despite referencing the signs of the end, Jesus teaches the sudden, unexpected taking away of those that were ready;
3) Luke 21:36: note that the "escape out from all these things" is associated with standing before the Son of man;
4) 2 Thessalonians 2:1-5: note that the Thessalonians had thought they missed it all, not only the gathering of the saints to Christ, but the coming of our Lord. The reference only makes sense if they were expecting a "secret rapture." Paul reassures them by reiterating that the coming will not be secret [even if the gathering is];
5) Revelation 3:7-13: note that there is a Jewish/Gentile divergence referenced and that the church in Philadelphia was promised to be kept out from the hour (a short period) of trial coming upon the whole earth;
6) Luke 17:26-36: Note that the rescue in the ark was followed by wrath on the earth, which makes perfect sense in light of 1 Thessalonians 5:9;
7) Revelation 12:1-6: this will actually take some words to develop, so please read on.
The woman is the messianic Jewish community, the baby in her womb is the Gentile church. The church is in the womb because it was hidden from sight from the former prophets (between Daniel's 69th and 70th week), even though the Jews were prophesied to have an affect on all nations. Though hidden, it grows and develops until it has attained its full number (remember Romans 11:25), at which point, gestation is complete and the baby born. Immediately the child is translated to the heavenlies (remember the church is the body of Christ and that we will rule and reign with him), but the Jewish mother is left, protected on earth for three and a half years.
There you have it. There is a sudden and escaping translation of the Gentile church to heaven at the turn from the Age of the Gentiles to the 70th Week of Daniel. Regardless of how the clandestine nature of this event has been described in the past, I guess we can say now, the secret's out!