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In java you can instantiate several instances of even correctly implemented singleton in case you are working with several classloaders.
Another possible answer is 'in distributed environments'.

When is Singleton not a singleton -
http java dot sun dot com/developer/technicalArticles/Programming/singletons/

imo, Singleton wont behave according to its definition, when we successfully Serialize the class object OR apply clone() on the class object. i think, both operations will clearly create new reference object.

Ps: Correct me, if i am wrong.

1, use reflection
2, multithread

when you incorrectly implement it??

that's a tricky one.
Lots of poor codes lead to that

when we serialization the object, the singleton may behave different

when we forget to throw cloneNotSupportedException,
to prevent the cloning of a singleton object

Possible scenories:
1) it should not work in a multi threaded enviornemnt...off course, not considered as double check locking mechanism...
2) If ctor of a class is in public mode

refer to dgmjava.blogspot.com

A singleton class will not work as Singleton in these situations
1. If not coded Thread safe
2. Across Process. Each JVM will have it's own Instance of Singleton
3. Using Multiple Class loaders can lead to multiple instances of same class
4. JVM before 1.2, Garbage collector will collect it if not maintained reference of Singleton

When you have public constructor.