This question makes two assumptions: first, that people are condemned for failing to believe in Jesus. This is false; though failure to believe in Jesus is most often sinful (exceptions would be infants, etc.), people are condemned for their sin in general and not for any sin in particular. According to Romans 6:23, the wages of sin is death. Ezekiel 18:4 reiterates the point: the soul that sins shall die. It is sin as such that warrants judgment, and not simply a certain kind of sin.
Second, the question assumes that there are sincere people who do not eventually become Christians. Jesus appears to indicate that this is false as well. In John 7:17 he says, “If anyone is willing to do His will, he will know of the teaching, whether it is of God or whether I speak from Myself.” Thus, if there are such people, we may justifiably infer that they will be brought to faith.
Of course, we realize that atheists do not accept the Bible as authoritative. However, the question above is aimed at showing that, granted the authority of the Bible, certain problems arise (i. e. people go to Hell because they lack a particular belief). What the biblical references attempt to show, then, is that this is in fact not the case. The Bible does not actually make the assertion assumed by the question.