There are many folklore tales of animals being used to predict weather. Just a few examples from the Old Farmers Almanac:
When cats sneeze, it is a sign of rain.
If the mole digs its hole 2½ feet deep, expect severe weather; if two feet deep, not so severe; if one foot deep, a mild winter.
Bats flying late in the evening indicates fair weather.
I have not been able to find much evidence to back these tales that I have come across. However, although not necessarily a prediction, the relationship of a cricket's chirp to temperature does seem to show how in tune animals can be with weather conditions. In fact, this relationship is also known as Dolbear's law, named after the American physicist Amos Dolbear, who published an article on the subject called The Cricket as a Thermometer. It is expressed as:
where Tf is degrees in Fahrenheit and N60 is chirps per minute
It is generally believed that Dolbear observed the snowy tree cricket to come up with his equation. The formula is believed to be accurate to within a degree or so for the field cricket. Generally speaking, the relationship is believed to hold true because as temperature rises, the cold blooded cricket's metabolism will also rise, providing more energy for muscle contractions and thus for chirping.
Are there observable behaviors to look for in animals that can be a harbinger of changing weather conditions? For example, say you are on a 2-3 day hike with no way to access weather forecasts and somehow you forgot to check the forecast before leaving.