Inner classes are appropriate when trying to emulate multiple inheritance. It is similar to what happens under the hood with C++: when you have multiple inheritance in C++, the object layout in memory is actually a concatenation of several object instances; the compiler then works out how the "this" pointer shall be adjusted when a method is invoked. In Java, there is no multiple inheritance, but an inner class can be used to provide a "view" of a given instance under another type.
Most of the time, it is possible to stick to single inheritance, but occasionally multiple inheritance would be the right tool to use, and this is the time to use an inner class.
This means that inner classes are somehow more complex than usual classes, in the same way that multiple inheritance is more complex than single inheritance: many programmers have some trouble wrapping their mind around that concept. Hence the "best practice": avoid inner classes because it confuses your coworkers. In my view, this is not a good argument, and at my workplace we are quite happy to use inner classes when we deem it appropriate.
(A minor drawback of inner classes is that they add one extra level of indentation in the source code. This is a bit irksome at times, when one wants to keep the code within 79 columns.)