1 Timothy 4:10

I recently heard a preacher quoting 1 Timothy 4:10 to argue that on the cross Christ saved the human race, but that those who will actually be saved are those who do not reject that salvation. Is that what the text is saying?

I suppose you have in mind the second part of 1 Timothy 4:10: "Because we trust in the living God, who is the Saviour of all men, specially of those that believe." The interpretation of this verse has been problematic, and consequently, different explanations have been given to the words of the apostle.
      If you carefully read any English translation of this passage, it does not seem to make sense. How can everybody be saved and at the same time have a group that is specially saved? What does "specially saved" mean? Does it mean that the others did not deserve to be saved and yet they are also going to be saved?
      When it comes to salvation, one is either saved or not saved. There is no middle ground; no one is going to be specially saved versus just saved.
      Several solutions have been proposed to answer these questions. One of them argues that the word Saviour here means "benefactor" in the sense that God provides for the needs of all people. This idea is usually rejected, because the term Saviour in the Pastoral Epistles points to the saving work of Christ in a very specific way.
      Another solution finds in this text the idea of universalism; that is, that God will ultimately save every human being. At the present time God is specially the Saviour of believers, but at the end He will show Himself to be the Saviour of all people; no one will be lost. Yet the New Testament again and again indicates that some people will be eternally lost.
      The interpretation that you heard is another attempt to explain this passage. It argues that on the cross God legally saved the world (He is the "Saviour of all men"), but that the individual is specially saved when that salvation is not rejected.
      There are several problems with this view. First, the idea that all have been saved is not found in the Pastoral Epistles. For instance, in 1 Timothy 2:4 Paul writes that God "desires all men to be saved and to come to the knowledge of the truth" (RSV). If the interpretation under discussion were right, one would expect Paul to say that God desires all persons to recognize that they have been saved. But Paul's statement indicates that not everyone has been saved in any form or way.
      Second, the idea of a legal salvation is not present in the text. It simply states that God is the Saviour of all, and this is considered by the apostle to be a fact.
      Third, in the King James Version we are using, the word "specially" seems to imply that "all men" are also actually saved; a view this interpretation ignores. There is no indication in the text that salvation is limited in any way to some people and not to others. I believe there is a better solution.
      The key term in the text is the word "specially" (Greek: málista). A proper understanding of it will solve the problems. The word can be translated "specially, mostly, above all." Recent studies made on the usage of this term in Greek papyrus letters have shown that this particle was used to define in a more precise way what preceded it and could be rendered "that is," "in other words," or similar expressions. If we use that translation in 1 Timothy 4:10, the interpretational problem disappears: "Because we trust in the living God, who is the Saviour of all men, that is, of those that believe." The phrase "all men" would then mean "all sorts of people," possibly Jews and Gentiles. There is no need to introduce the idea of universalism or of a legal salvation that has no immediate effect on the final destiny of the individual.
      On the cross God made provision for the salvation of every human being, but only those who believe will be saved. Christ's substitutionary atonement is universal in its extent, but because of the freedom God has given to humans, it is limited in its effect on our salvation.