Q. What is the significance of a Shabbat candle that went out three times? - Revital, Petach-Tikva
A. Shalom Revital. Generally, the explanation is that the candle is deficient, or that there is a wind in the place where you light. Therefore, you should change either the candle or the place where you light.
Meanwhile, Rabbi Professor Jeffrey Woolf has a terrific post with a story about the Kovner Rav and a story about Rav Gedalyah Felder. Read them at this link.