The original Bible words that have been translated to the English word saint are found through the entire Old and New Testaments. Saints did not originate at the time of the earthly life of Jesus Christ.
All of the original words above (listed in their transliterated form - written according to how they sound in English) all carry the same definition of a saint - someone who is sacred, holy, pure, blameless, dedicated. They became saints by means of the Holy Spirit, which can only come from God. God therefore chooses His saints, and gives them of His Holy Spirit to make it possible (see The Elect).
The people described as saints in The Bible were however still very much human. They were called, they were holy, and they were extremely dedicated (both in terms of attitude, and in the sense of being set apart), but they were still real people, far from perfect. Saints never stopped being normal people - fishermen, farmers, tent makers, doctors, teachers, carpenters. The "little people" of the congregations were as much saints as the most famous and prominent ones such as Peter and Paul (e.g. Romans 15:26).
Saints were not infallible (e.g. Galatians 2:11), and sometimes they disagreed with one another (e.g. Acts 15:39). They were most certainly called, converted, good people, but they did not become somehow perfect. As far as The Bible is concerned, God's Word was and is perfect, but God's people, while yet physical, were and are not perfect in knowledge, or behavior (see Paul's Mirror).
Were there more saints after those recorded in the Bible? Yes. Have there been true saints during the past 2,000 years? Yes, secular history records their persecution, along with the persecution of millions of others who were "saints" of various man-made church organizations. Are there true saints alive on the earth today? Yes, they and/or their descendants are described in end-time prophecy.
Bible Prophecy has much to say about God's true saints living on the earth during the end-times. Circumstances will be extremely difficult, but they will eventually triumph. A few examples from your Bible:
"If any one is to be taken captive, to captivity he goes; if any one slays with the sword, with the sword must he be slain. Here is a call for the endurance and faith of the saints." (Revelation 13:10 RSV)
"Here is a call for the endurance of the saints, those who keep the Commandments of God and the faith of Jesus." (Revelation 14:12 RSV) (see The Ten Commandments Now?)
"The third angel poured his bowl into the rivers and the fountains of water, and they became blood. And I heard the angel of water say, "Just art thou in these thy judgments, thou who art and wast, O Holy One. For men have shed the blood of saints and prophets, and thou hast given them blood to drink. It is their due!" (Revelation 16:4-6 RSV) (see The Seven Seals)
"and on her forehead was written a name of mystery: "Babylon the great, mother of harlots and of earth's abominations." And I saw the woman, drunk with the blood of the saints and the blood of the martyrs of Jesus. When I saw her I marveled greatly." (Revelation 17:5-6 RSV)
"Then I heard what seemed to be the voice of a great multitude, like the sound of many waters and like the sound of mighty thunderpeals, crying, "Hallelujah! For The Lord our God the Almighty reigns. Let us rejoice and exult and give Him the glory, for the marriage of the Lamb has come, and His Bride has made herself ready; it was granted her to be clothed with fine linen, bright and pure" - for the fine linen is the righteous deeds of the saints. And the angel said to me, "Write this: Blessed are those who are invited to the marriage supper of the Lamb." And he said to me, "These are true words of God." (Revelation 19:6-9 RSV)
Fact Finder: Who will be resurrected, or changed to spirit if alive then, to meet Jesus Christ at His Return?
1 Thessalonians 3:13, 4:16-17
See The Return Of Jesus Christ and The Last Day