When NOT to use the static keyword in Java?

One reason why you may not want it to be static is to allow it to be overridden in a subclass. In other words, the behaviour may not depend on the data within the object, but on the exact type of the object. For example, you might have a general collection type, with an isReadOnly property which would return false in always-mutable collections, true in always-immutable collections, and depend on instance variables in others.

However, this is quite rare in my experience - and should usually be explicitly specified for clarity. Normally I'd make a method which doesn't depend on any object state static.