When the humidity suddenly drops to a low level several things can occur:
The pegs can shrink and become loose, making it difficult to keep the strings in tune. This should be taken as a warning sign, and you should expose the instrument to more humidity as soon as possible. Of course, pegs can also slip if they don't fit well or are overly lubricated. If humidity increases dramatically, pegs can become too tight to tune properly. See Pegs for remedy.
Shrinkage can cause the top or back to separate from the sides in some areas. In this case, the shrinking action is stronger than the glue, and the top separates from the sides in order to accommodate the shrinkage. This does not harm the instrument and is relatively inexpensive to repair.
If the top or back does not separate from the sides to allow for shrinkage, the wood can crack. The repair of a crack is very costly.
The fingerboard can rise up closer to the strings, especially on cellos. When this happens, the strings might not have enough clearance to vibrate without hitting the fingerboard. In areas where humidity levels vary considerably, cellists often own bridges of different heights in order to compensate for the variation in fingerboard height caused by the humidity.