Originally, they would accept testimony for the New Moon throughout the day. Once, the witnesses were delayed and this caused the Leviim to sing the wrong song (this will be explained). They decreed that they would only accept witnesses until Minchah time. After that, they would treat both that day and the following day as Yom Tov.

After the Temple was destroyed, R. Yochanan B''Z decreed that they should again accept witnesses all day.

(Gemara - R. Shmuel bar Nachman): The reason for the witnesses' delay is - Once, a fire broke out in the city and gentiles came to loot the area. They kidnapped the witnesses and by the time they were freed, they came late to the Temple and the Leviim had already sung the weekday song.

R. Acha bar Papa to R. Zeira: Chevraya asked R. Shmuel bar Nachman - in whose name did you make this statement?

R. Shmuel B''N answered - Am I like you, that you have many Rabbis? I only learned Torah from R. Yehoshua ben Levi, and it is he who I quoted.

R. Zeira: R. Ila explained the reasoning behind the Mishnah as follows - In the Chapter recited on a Thursday (weekday), the Pasuk states (Tehilim 81, 2), "Sing to the G-d of our might''. On Rosh Hashanah (Yom Tov) they say - from the same chapter - (Tehilim 81, 7), "I removed his shoulder from the load''.

If Yom Tov occurs on a Thursday - in Shacharis they say (Tehilim 81, 2), "Sing to the G-d of our might''. In Minchah they say (Tehilim 81, 7), "I removed his shoulder from the load''. If they came after Minchah, they say "Sing to the G-d of our might''. This means that they have repeated the same song twice!

R. Ada of Kisarin quoting R. Yochanan: The reason that the Mishnah said that if the witnesses came after Minchah they would treat both that day and the following day as Yom Tov, is because most of the day had passed being treated as Yom Tov.

R. Chiya bar Ba: R. Yochanan ordered the Synagogues in Kifra that if the witnesses come after Minchah, as soon as the Beis Din begin to converse with the witnesses, even though they will treat the following day as Yom Tov, they should insert "Yaaleh VeYavoh'' into their prayers today. (It seems that we can be referring here to any Rosh Chodesh.)

(Mishnah): R. Yehoshua ben Karchah - R. Yochanan B''Z also decreed that even if the Head of the Beis Din moved elsewhere, the witnesses should go to the seat of the Beis Din itself.

(Gemara): The Mishnah is referring to the place where the Beis Din usually convene for sanctification of the new moon.