In season six episode five, "Home," the Night King was able to touch Bran, marking him with an icy palm print. This enabled the Night King to break the magic of Bloodraven's cave.

If some theories are correct, it raises questions about the safety of Westeros, too. We know the Wall has magic weaved into its foundation (Benjen said as much in the season six finale), and that magic is probably the same as the magic that protected Bloodraven's cave.

Bran passing through the Wall might somehow enable the Night's King to destroy the Wall and enter Westeros. That would be bad news for everyone.