The idea of people being left behind seems to come from Matthew 24:
37 For as were the days of Noah, so will be the coming of the Son of Man. 38 For as in those days before the flood they were eating and drinking, marrying and giving in marriage, until the day when Noah entered the ark, 39 and they were unaware until the flood came and swept them all away, so will be the coming of the Son of Man. 40 Then two men will be in the field; one will be taken and one left. 41 Two women will be grinding at the mill; one will be taken and one left.
It seems to me, though, that interpreting this text as the rescue of believers from earthly corruption almost totally reverses the meaning of the passage. The comparison with the days of Noah makes this plain. In the days of Noah, it was the ungodly who were taken away, experiencing God's wrath in judgment for their sin. To be left behind -- as Noah and his family were -- was to experience God's mercy and grace.
So who is it that's going to be left behind?