Motherboard has a great story (with no New Hampshire connection, sorry, but I liked it) concerning the way two Supreme Court justices differ on whether to capitalize the word “internet” (or “Internet”):

These differing opinions arise from fundamentally different understandings of the Internet, which coincide with the arguments for and against capitalizing the word. Proponents of uppercase-Internet contend that the Internet is a singular place, like the Earth, making it a proper noun. Those who favor lowercase argue that the internet is a generic tool, like earth, of interest primarily for what can be built with and upon it, and too common and ubiquitous to merit a capital letter.

Whether the Internet is a single place or just a way to reach many places—that is at the heart of the differing views.

Kennedy, who consistently capitalizes “Internet,” describes the system “in a spatial sense” and treats it as a single, massive conversation for “knowing current events, checking ads for employment, speaking and listening in the modern public square, and otherwise exploring the fast realms of human thought and knowledge.”

In response, Alito, who exclusively uses lowercase-internet fifteen times, chides Kennedy for “musings that seem to equate the entirety of the internet with public streets and parks.” He scare-quotes a reference to “a ‘place’ ( i.e., the internet),” and his analysis ultimately comes down to analyzing whether individual internet websites raise First Amendment concerns. The genericism of little-i internet amplifies Alito’s dismissal of the Internet as a single, unified dialogue, in favor of a disaggregated internet that can be restricted or blocked in parts

 You can – and should -read the whole piece here.

I used Internet, the original AP style,  for years and still have trouble not capitalizing it.