It was a Tuesday, and I called to make a medical appointment. I was told that one was available "next Thursday". Being a literalist by nature, my first thought was "an opening in just two days time?". Something about the unlikelihood of this made me check. It was "Thursday next week" that was being offered to me.
My guess is that most Americans consider "next" in the context of dates to mean "next week". Within the week, the next occurrence of a given day of the week is referred to simply as (e.g.) "on Thursday".
Well, I got that right, and set up my appointment for Thursday of the following week.