The defendant in Bronston v. United States, 409 U.S. 352 told a partial truth but not the whole truth, and got away with it. The crucial feature of his testimony is that he only answered part of the question asked, and it was the duty of the opposing lawyer to detect the discrepancy. This gave rise to the "literal truth" standard, which has then been clarified in various ruling such as US v. De Zarn which focus on the entire context of testimony, not just on the absolute truth value of an isolated sentence.
The problem with the concept "the whole truth" is that it is impossible to testify to everything that you know to be true, you can only testify to that which you know to be true and relevant. The form of the question can dictate what is relevant, for example if the question asks "did you see Smith on the night of June 30", then not mentioning anything about June 29 or the morning of June 30 would not be "failing to tell the whole truth", and it would not be perjury if you answered "No" even when you say Smith on the morning of June 30. If the question was more broadly "Have you ever seen Smith", then in those circumstances, "No" would be perjury. In other worth, you are entitled to assume a certain scope of relevance, if not clearly contradicted by the whole of the testimony (which is why you need a lawyer to watch out for your interests).
If, ill-advisedly, you say "No." intending to continue "Well, actually, yes, I did see him on the morning of the 30th", you might get dismissed by the attorney. Before you leave the stand, you can address the court and ask permission to correct your testimony.