Two passages of scripture come to mind to make me doubt that there was a thirteenth month.

I Chronicles 27:1-15
This is where David assigns chiefs for each month of the year. He appoints just twelve; one for each month of the year without considering a thirteenth month.

Revelation 22:2
This talks about the tree of life that yields twelve manner of fruits each month. It does not say thirteen.

Jews have been using a lunisolar calendar since Biblical times, but originally referred to the months by number rather than name. Only four pre-exilic month names appear in the Tanakh (the Hebrew Bible): Abib (first, literally "Spring"), Ziv (second), Ethanim (seventh), and Bul (eighth), and all are Canaanite names, and at least two are also Phoenician. It is possible that all of the months were initially identifiable by native Jewish numbers or foreign Canaanite/Phoenician names, but other names do not appear in the Bible.

Furthermore, because solar years cannot be divided evenly into lunar months, an extra "embolismic" month must be added to prevent the starting date of the lunar cycles from "drifting" away from the Spring, although there is no mention of this in the Bible. In Second Temple times, the beginning of each lunar month was decided by two eyewitnesses testifying to having seen the new crescent moon. Patriarch Gamaliel II (c. 100) compared these accounts to drawings of the lunar phases. According to tradition, these observations were compared against calculations made by the main Jewish court, the Sanhedrin. Whether or not an embolismic month (a second Adar) was needed depended on the condition of roads used by families to come to Jerusalem for Passover, on an adequate number of lambs which were to be sacrificed at the Temple, and on the earing of barley needed for first fruits.

Once decided, the beginning of each Hebrew month was first announced to other communities by signal fires lit on mountaintops, but after the Samaritans and Boethusaeans began to light false fires, special messengers were used. The inability of the messengers to reach communities outside Israel within one day, led outlying communities to celebrate scriptural festivals for two days rather than for one, observing the "second feast-day of the Jewish diaspora."

From the times of the Amoraim (third to fifth centuries), calculations were increasingly used, for example by Samuel the astronomer, who stated during the first half of the third century that the year contained 365 ¼ days, and by "calculators of the calendar" circa 300. Jose, an Amora who lived during the second half of the fourth century, stated that the feast of Purim, 14 Adar, could not fall on a Sabbath nor a Monday, lest 10 Tishri (Yom Kippur) fall on a Friday or a Sunday. This indicates a fixed number of days in all months from Adar to Elul, also implying that the extra month was already a second Adar added before the regular Adar.

In leap years, a 30 day month called Adar I is inserted immediately after the month of Shevat, and the regular 29 day month of Adar is called Adar II. This is done to ensure that the months of the Jewish calendar always fall in roughly the same seasons of the solar year, and in particular that Nissan is always in spring. Whether either Chesvan or Kislev both have 29 days, or both have 30 days, or one has 29 days and the other 30 days depends upon the number of days needed in each year. Thus a leap year of 13 months has an average length of 383½ days, so for this reason alone sometimes a leap year needs 383 and sometimes 384 days. Additionally, adjustments are needed to ensure certain holy days and festivals do or do not fall on certain days of the week in the coming year. For example, Yom Kippur, on which no work can be done, can never fall on Friday because the high fast could not be broken at sunset — because the end of Yom Kippur would be the start of the Sabbath, on which no work can be done. Thus some flexibility has been built in.

The 265 days from the first day of the 29 day month of Adar (i.e. the twelfth month, but the thirteenth month, Adar II, in leap years) and ending with the 29th day of Heshvan forms a fixed length period that has all of the festivals specified in the Bible, such as Pesach (Nissan 15), Shavuot (Sivan 6), Rosh Hashana (Tishri 1), Yom Kippur (Tishri 10), Sukkot (Tishri 15), and Shemini Atzeret (Tishri 22).

The festival period from Pesach up to and including Shemini Atzeret is exactly 185 days long. The time from the traditional day of the vernal equinox up to and including the traditional day of the autumnal equinox is also exactly 185 days long. This has caused some unfounded speculation that Pesach should be March 21, and Shemini Atzeret should be September 21, which are the traditional days for the equinoxes. Just as the Hebrew day starts at sunset, the Hebrew year starts in the Autumn (Rosh Hashanah), although the mismatch of solar and lunar years will eventually move it to another season if the calendar isn't reformed (this will not happen for thousands of years). (Source: Wikipedia)