Torah and Shavuot
Was the Torah really given on Shavuot? 
This cannot be definitely proved since Shavuot falls on different days of the year, depending upon which day of the week Pesach begins. The claim is very reasonable, however.
“In the third month after the people of Isra'el had left the land of Egypt, the same day, they came to the Sinai Desert. After setting out from Refidim and arriving at the Sinai Desert, they set up camp in the desert; there in front of the mountain, Isra'el set up camp. (Exodus 19:1-2, Complete Jewish Bible)”
The phrase “the same day” is interpreted as the first day of the third month, Sivan. On the first day of Sivan they came to the Desert of Sinai and camped at Refidim, or Rephidim in other translations. It is a day's journey from Refidim to Mt. Sinai, so they arrived at the mountain on the second day. On the third day, Moshe would have gone before the Lord.
“…Adonai said to Moshe, "Go to the people; today and tomorrow separate them for me by having them wash their clothing; and prepare for the third day. For on the third day, Adonai will come down on Mount Sinai before the eyes of all the people. (Exodus 19:10, CJB)”
Three days of consecration brings us to the fifth day of Sivan. Shavuot can happen anywhere from the fifth to the 12th day of the Sivan.
More about Shavuot