I'm currently working on a bigger post about switching my family to the iTunes Family Sharing plan for the second time. Things are going much more smoothly this time than the first attempt but there are still some interesting points to discuss. In the meantime, I wanted to share this one bit of information that took some sleuthing to figure out.
With iTunes Family Sharing, every member of the family is his or her own island, but not entirely. Specifically, the five computer limit in iTunes is spread across all of the accounts attached to your Family Sharing plan. Each person doesn't have the option of connecting five Macs. Instead you have five Macs to be distributed among the whole group. If you've got more than five computers and are running into authorization errors, that is probably the reason.