What is the meaning of calling a method or object "thread-safe?"
For an object to be thread-safe, it must be possible for multiple threads to simultaneously access the same method, or multiple methods, in that object. Usually this is acheived by assuring that each method is thread-safe, but this doesn't always suffice, since methods can call each other in strange ways, leading to deadlock and other weirdness.
It is very difficult to prove that an object is thread-safe. The main rule of thumb for making thread-safe objects is, "Make all the instance variables private, and all the public accessor methods synchronized." However, this is sometimes difficult to achieve in practice, due to exigencies of performance, architecture, or implementation.