When was the chain of Semicha from Mt. Sinai broken?
Originally, Semicha was a direct line from Moshe, but when did it break?
The exact date that the original semikhah succession ended is not certain. Many medieval authorities believed that this occurred during the reign of Hillel II, around the year 360 CE. However, Theodosius I forbade the Sanhedrin to assemble and declared ordination illegal. (Roman law prescribed capital punishment for any Rabbi who received ordination and complete destruction of the town where the ordination occurred). It seems to have continued until at least 425, when Theodosius II executed Gamaliel VI and suppressed the Patriarchate and Sanhedrin
The Rebbe (Lubavitch) discusses this in a few places. See for example Sefer hasichos 5752 page 230. בארוכה בשיחת ש"פ בראשית תשמ"ה (התוועדויות תשמ"ה ח"א עמ' 414 ואילך) (וקודם לזה, בקיצור, בשיחת ער"ה תשד"מ (התוועדויות תשד"מ ח"ד עמ' 2683 ואילך). ולהעיר משיחת יום שמח"ת תשי"ט.
An argument can be made that the chain of semicha effectively ended with the destruction of the first (Solomon's) temple and the exile of the Jews into Babylon. Arcahelogical evidence reveal that Jerusalem was largely destroyed, with the entired walled city being burnt to the ground. There is some evidence that other areas of Judah were not deported. However, the temple, which was the center of Jewish religious life, was lost and the Jews were no longer in political power. The Babylonian captivity lasted for 70 years as prophesized by Jeremiah, and it ended in 538 BCE when Cyrus of Persia issued a general decree allowing exiled people to their lands.
The book of Ezra indicates that when the Jews returned to Israel, Ezra had to reintroduce the Torah to the people (Ezra Chapters 7-10 and Nehemiah Chapter 8). Many Jewish men had taken wives from the nations. It also seems clear that most Jews were no longer aware of many of the holidays. As further evidence of the influence of the captivity, consider that certain names for the months were adopted from Babylonian (e.g. תַּמּוּז , מַרְחֶשְׁוָן). Tanach also does not make mention that the Jews who had stayed behind had been observant for the two generations of the exile. Therefore, as all Jews had become largely unobservant during the exile, it seems plausible that the right of semicha could easily have fallen into disuse. Even if semicha were continued in Bablyon or Judah, the argument could be made that the transfer were invalid since either party may not have been observant (similar to the modern halacha rejecting a reform ordination).