In fact, the chances are considerably better than that. Several years ago I stumbled across Luke 1:26 out of context and really noticed for the first time what it might be saying:
And in the sixth month, the angel Gabriel was sent from God into a city of Galilee, called Nazareth....That is the Douay-Rheims version, and it is rather literal, but several modern translations tell you not what the text actually says, but what the translators think it means. For example, the New Living Translation starts with, "In the sixth month of Elizabeth's pregnancy...", the International Standard Version begins with, "Now in the sixth month of her pregnancy...", and the God's Word Translation has, "Six months after Elizabeth had become pregnant...." What if, though, the "sixth month" does not refer (or does not exclusively refer) to Elizabeth's pregnancy, but (also) to the Jewish calendar?
Today we have several different calendar years; we have civil years, fiscal years, and academic years, and in the Catholic Church the year starts with the first Sunday of Advent, not January 1. In the same way the Jews of the first century had both a civil year (that begins with Rosh Hashanah) and a religious year, but the default meaning for "Jewish New Year" seems to be Rosh Hashanah. If we take six months from Rosh Hashanah, then add nine more for Mary's pregnancy, we get three synodic months of 29.5 days from the end of Rosh Hashanah for the date of Christmas. The actual date fluctuates because the Jewish calendar is a lunar calendar, but Rosh Hashanah tends to be in mid-to-late September, yielding dates for Christmas that are within a couple of weeks of December 25.
More importantly, this same trend works for the first few years B.C.. To get the Julian date of Christmas we have to subtract 4 days due to the difference between a synodic month and our months of September, October, and November, and another two days from the Rosh Hashanah date to convert to the Julian date from the Gregorian date. This means that the earliest possible Christmas dates by this method are as follows: 1 A.D. -- December 10; 1 B.C. -- November 21; 2 B.C. -- December 2; 3 B.C. -- December 14; 4 B.C. -- November 24; 5 B.C. -- December 5. This covers at least the years when most people think Jesus may have actually been born.
Does this idea have any ancient support? Well, I have found nothing relating Christmas to Rosh Hashanah. However, St. Bede claimed (I have lost the precise reference!) that when Zachary went into the Temple to offer incense it was Yom Kippur, the Day of Atonement. That would push the "earliest dates for Christmas" back about 10 days, making for an even better fit with our December 25 observation of Christmas.