I used to teach this verse as simply making a logical argument that grace and works could not both be causes for salvation. They were mutually exclusive as causes. If salvation was by one, then it could not be by the other.
I still believe that this verse teaches this. However, a further study of the context has convinced me that there is a historical element to the "no more of works" phrase. Romans 11:1 declares the principle that God has not cast away His people, the Jews. Romans 11:2-4 gives the example of the time that Elijah thought he was the only true believer left in Israel when there were 7,000 who had not bowed to Baal. Romans 11:5 assures the Romans that "at this present time also there is a remnant according to the election of grace." Then, Romans 11:6 declares that the present remnant survives "no more of works," but now comes by grace.
The problem in interpretation comes because some Bible teachers assume that the comparison is of ways to receive eternal salvation. Although salvation is certainly included in those who are presently in the remnant according to grace, this is not the theme of this passage. The passage specifically refers to the preservation of a remnant among God's chosen earthly people--the Jews. Under the law, which was definitely a covenant of works, they maintained their status as the people of God by their works. In this age, they can truly be the people of God only by the election of grace--that is, by eternal salvation.
However, since the Bible is clear that the keeping of the law could save no one (Acts 13:39; Romans 3:20; Galatians 3:21-22; Hebrews 7:11, 19; 9:9), being part of the Old Testament Jewish remnant was not identical with eternal salvation. The text deals with being God's earthly people. In the Old Testament, this was maintained by the works of the law. In the New Testament, it is obtained by the election of grace through Jesus Christ. However, in the Old Testament, being God's faithful earthly remnant was not identical with having eternal salvation. Certainly, there were Gentiles who knew God as personal Saviour who were not a part of the Jewish remnant. Also, there were probably Jews who did not bow to Baal who had never truly believed in God as Saviour.
Romans 11:1-6 does indicate a chronological difference, but it is a difference in how God preserved His remnant among the children of Israel. It was not a difference in how people went to heaven.
You also mention Romans 3:21, which states, "But now the righteousness of God without the law is manifested, being witnessed by the law and the prophets." There was a "righteousness of the law" (Romans 2:26) that was the result of keeping the law. Yet, however praiseworthy this righteousness was, it was not equivalent to eternal salvation. In fact, a legalistic keeping of the law was not a way to salvation.
Romans 9:31-32 states, "But Israel, which followed after the law of righteousness, hath not attained to the law of righteousness. Wherefore? Because they sought it not by faith, but as it were by the works of the law. For they stumbled at that stumblingstone." Here we see that faith was the key to true righteousness even under the law. External righteousness could be maintained without a true internal salvation. Those who tried to establish their own righteousness through the law (Romans 10:3) would not find God's true righteousness.
Therefore, Romans 3:21 is not dealing with two kinds of salvation but rather with two kinds of righteousness: the righteousness of the law that could not save and the righteousness of Jesus Christ by which we are saved.
