Some North Americans often say the following:
IN-ah-net (Internet)
moun-IN (mountain)
IN-ah-res-ted (interested)
IN-ah-national (international)

I think this is called a "glottal stop". Is this characteristic of any particular region of the States?

Harumph! Those pronunciations are just careless...and for those of us who are deaf or hard of hearing they are frustrating.
What ever happened to enunciation? Harumph!
The imprecise pronunciation is not a glottal stop. A GS is a specific intonation most often heard in US in Hawai'i where the stop is annotated by a backwards apostrope which is not on my keyboard.
It looks like this Liliu'okalani - the name of Hawai'i's last queen...pronounced Lily-oo(stop)oh-ka-lani with the accent on the oh ka. [The oo like the end of you !]