There is no call in The Bible to try and predict in advance when various days will fall.
The first of Nisan is the only new year that is Biblical. And it should always be the first New Moon after the Barley Harvest in Israel starts. When the Barley Harvest happens in Adar (or second Adar if they have one) then the Rabbinic calendar happens to be right. And the Karaite reckoning does wind up corresponding with the Rabbinic calendar more often the Samaritan Calendar does.
If the Barley Harvest happens late and i doesn't occur in Adar, then that is when you do a second Adar, whether or not it was expected. And if the Barley harvest happens in first Adar, then there shouldn't be a second Adar whether or not the Rabbinics had planned one.
Likewise for the months. The Rabbinics have complicated way of counting things, and because of that even when the month is right Tabernacles often doesn't fall on the true Astronomical Full Moon. If the New Moon happens sooner then expected then that months ends early, if it happens latter then expected then that months has an extra day or two.
Now today we can predicts New Moons and Full Moon pretty accurately with our astronomical sciences that the Ancient Hebrews didn't have. How nature cooperates for the Barley Harvest is not as dependable however.
Christians who think The Biblical calendar still has significance, should start as soon as it's February looking for the Barley Harvest in Israel.
Karaites aren't Christians, but they are the Protestants of the Jewish world in terms of Sola Sicprtura. Some may at times be like many Protestants who pay lip service to Sola Sciprtura while attacking Catholics but then just build traditions of their own. But it looks from the website I linked to that they have a pretty solid understanding of the Biblical reckoning of the times and seasons.
Update Jan2016: Knowing that the Rabbis are doing can still help predict some basic things, the New Moons are not likely be off by any more then 2 days. And from there using Stellarium from a view point in Israel can maybe help one predict if the New Moons will be off. The Rabbis are expecting 2016 to have a Second Adar. And with Shevet beginning less then half way into January I think it's likely there will be one for the Karaites too.
Some Karaites have chosen to outright identify themselves with the Sadducees and deny the Resurrection of the Dead The Afterlife and a coming Messiah. But not all take that position.