English Explanation of Mishnah Sanhedrin 3:8

So long as a litigant can produce proof he may overturn the verdict.
If they had said to him, “Bring all of the proofs that you have within thirty days” and he brought them within thirty days, the court may overturn the verdict. But if he brought any proof after thirty days, the court cannot reverse the verdict. Rabban Shimon ben Gamaliel said: “What could he have done that he did not find [the proof] within thirty days but found it after thirty days?”
If they had said to him, “Bring witnesses” and he said, “I have no witnesses”, or [if they said], “Bring proof”, and he said, “I have no proof”, and he later found proof or witnesses, then they are totally invalid. Rabban Shimon ben Gamaliel said: “What could he have done that he did not know that he had witnesses, then found witnesses, or that he did not know that he had proof, then found proof?
If they had said to him, “Bring witnesses” and he said, “I have no witnesses”, or [if they said], “Bring proof”, and he said, “I have no proof”, but when he saw that he was about to be found obligated, he said, “Come near, so-and-so and so-and-so and testify for me!”, or if he brought forth some proof from his wallet, then they are totally invalid.
Mishnah eight discusses the ability of the litigants to overturn the verdict by bringing new evidence (physical proof or witnesses) after the trial.
Mishnah eight discusses the situation where a person did not bring proof or witnesses until after he was found to be obligated. If the court had not told him that he had a time limit on bringing such evidence, he may always bring it and thereby give the judges reason to overturn the verdict. If, however, the court had set a time limit on his bringing proof or witnesses, and that time limit elapsed, he may no longer attempt to overturn the verdict. Since the court set a time limit, anything brought afterwards is not considered relevant to the trial. Rabban Shimon ben Gamaliel disagrees with this time limit. According to him even if a time limit was set, if the defendant should find new evidence he may bring it to the court’s attention and have the decision overturned. In other words there is a dispute between Rabban Shimon ben Gamaliel and the Sages (the anonymous opinion), the former holding that time limits are irrelevant and the latter that they are valid.
The final clause of the mishnah contains an opinion to which even Rabban Shimon ben Gamaliel agrees. Even though he generally allows evidence to be brought later, if it is clear that the defendant could have known about the evidence and chose not to use it and then produced it just at the moment when the verdict was about to be delivered against him, the evidence is invalid. In such a case we may rightfully wonder why he didn’t bring the evidence earlier and we suspect him of lying.