Most dates I work with are near the present day. Scheduling something next week, referring to last week. Very rarely do I exceed the current month unless I am near a month boundary. The current month is nearly always known information.
In this case, the day is the first piece of information I look for (and almost always when I want to know today's date, all I am really asking for is "what day of the month are we on today?").
And (as a European) I'm constantly amazed by how weird and confusing it is to put the middle-level information up front. If there were different separators (e.g. if today were 12,16/2013) I think I'd struggle less.
11/02/13 12/03/13 15/02/13 14/03/13
So disappointing when you realize the apartment isn't available until next month.