Too humid: Wood swells; pegs stick tight. This can cause pegbox cracks.
Glue absorbs moisture and seams may open. Too dry: Wood shrinks; the pegs slip. In extreme cases the instrument may seem to fly apart; the strings loosen, the bridge fall down, and even the soundpost may fall. The saddle shrinks less than the top will, which can cause cracks on the face of the instrument.
Too warm: Glue softens, causing open seams. (The worst scenario is to leave your violin in the trunk of your car. It’s possible to come back to a caseful of violin parts.) Varnish softens and sticks to the case. Wood swells. The humidity is usually higher when the temperature is higher, making things worse.
Too cold: Varnish shrinks and can cause craquelure (fine cracks). Glue becomes brittle; seams may open. Pegs shrink, loosening the strings.
Recommendation: Keep your instrument where you would feel comfortable. Room temperature and 50% humidity is best. Minimize changes in temperature and humidity. For instance, when you come inside in the winter, unzipper the case but do not open it for fifteen minutes. This allows the instrument to gradually warm up and the moister air to infiltrate gradually.