As some others said, many times, when you use an anonymous inner class, it is also used on some other places too...
Thus you may easily duplicate inner class code to many places...
This seems not a problem when you are using very simple inner classes to filter/sort collections, using predicates, comparator or anything like that...
But you must know that when you use 3 times an anonymous innerclass that does exactly the same thing (for exemple removing the "" of a Collection), you are actually creating 3 new classes on the java PermGen.
So if everyone use inner classes everywhere, this may lead to an application having a bigger permgen. According to the application this may be a problem... If you are working on the industry, you may program embedded applications that have a limited memory, that should be optimized...
Note this is also why the double curly brace syntax (anonymous innerclass with non-static initialization block) is sometimes considered as an antipattern:
new ArrayList<String>() {{
add("java");
add("jsp");
add("servlets");
}}
You should ask to people who forbids you to use them...
IMHO it all depends on the context...