1. It is a Mitzvah for the new moon to be sanctified by Beis Din through the testimony of witnesses.
2. The Gemara discusses whether the Adar that is next to Nisan is always a partial month (29 days).
3. There is a dispute about why an extra day was made for Elul during the days of Ula.
4. Even if witnesses came on the thirtieth day of a month, it was still possible to make the month a partial month.
5. If Beis Din needed to extend the month, they would try to ensure that valid testimony was not presented earlier.
A BIT MORE
1. According to one explanation, this is why witnesses who see the new moon of Nisan should even transgress Shabbos to come to testify before Beis Din immediately, even though we know when Rosh Chodesh is going to be (since every Adar that is next to Nisan is only a partial month of 29 days).
2. The Gemara concludes that it is sometimes a full month (30 days) and sometimes it is a partial month (29 days).
3. Ula: This was done to prevent Shabbos and Yom Tov from being consecutive. When they are consecutive, the fresh vegetables that were picked for Shabbos and Yom Tov tend to wilt and become unappetizing on the second day. Rebbi Acha bar Chanina: This was done in order to prevent a person who died before two consecutive days of Shabbos and Yom Tov from rotting for two days before being buried.
4. If there was a need to do so, Beis Din would extend the month despite the testimony of the witnesses.
5. One of the tactics used was to question any witnesses who would come early (on the thirtieth day) at great length in order to get them to err in their testimony, to render it invalid.