Probably not, since the hills around Bethlehem are not prone to receiving snowfall.
Bethlehem has a Mediterranean climate, with hot and dry summers and cold winters. Winter temperatures (mid-December to mid-March) can be cold and rainy. January is the coldest month, with temperatures ranging from 1 to 13 degree Celsius (33-55 °F) - a little warm for snow, but rather cold for shepherds to be out in the fields at night.
When you couple that with the probability that the Nativity occurred NOT in mid-Winter but more likely the spring around early April (when sheep were first taken out to the fields after Winter and are lambing - hence the shepherds watching over their flocks by night, to assist them in the birthing process), and it is highly unlikely there would be any snow around. The current 25 December date for celebrating Christmas was not adopted until much later in the Christian era and is widely recognized as being something of a compromise to allow the pagans who converted to Christianity to continue celebrating their mid-Winter holidays.
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Luckily, for the 3 wise men, no. We are used to Christmas occuring during the winter seasons, but we've adopted different calenders (Julius Ceasar's, Ramses II) as well as created daylight savings time. Because of this, our months have shifted around the seasons. The first Christmas didn't have any of this and would most likely have been during September. Hope this helps! :)