I’m from Long Island
We pronounce it kuh-NET-uh-cut
Try living in Cuyahoga County; even people who’ve lived here their entire lives can’t agree on its pronunciation.
Well, it gave me entree for one of my favorite quotes. 
koNEteekah (I think I got that right in English phonics). Or, in Spanish phonics, konética
/kə.ˈnɛt.ə.ˌkɪt/
And pity the poor colonists who lived in quiquichtiqok. Thank god they renamed it Mystic 
Was the Connecticut spelling originally from an English-speaking person? I know that many of the early colonies were inhabited or influenced by French, Dutch and German immigrants. I tried doing a search on the etymology, but no one goes any further than citing the Algonquin origin. I love to blame the French for these kinds of things; maybe it was an English adoption of a French spelling for an Algonquin word?
I came in here to mention this very movie. It’s the only time I’ve ever heard the middle C pronounced.
How can you talk about Mackinac and Cheboygan without mentioning Sault Ste. Marie? I defy anybody who hasn’t been told to properly suss the pronunciation of that one.
“Soo Saint Marie”
Like most annoying things in the world, this is the fault (“foo?”) of the French.
WAG: You can find some early spellings with an “h” in the plact of the middle “c.” My guess is that in the original Indian language, that “h” was actually a rendition of some aspiration or breathy sound before the “t,” but some speakers ignored it completely. The middle “c” survived as an artifact in the formalized spelling, but the pronunciation by this time had already been established. Again, just a WAG.
Note that “sault” isn’t pronounced “soo” in French, but rather “so”.
A lot of the old timers in Colorado call the state ‘Ar-kan-saw’ and the river ‘Ark-ans-as’. It seems to be dying out though and settling on The state pronunciation for both.
In the movie Holiday Inn, Fred Astaire’s manager attempts to find out from a concierge where Fred Astaire has gone. Told that he has said he was headed to Connecticut, the manager asks why he didn’t stop Fred Astaire. The response: “How could I stop him when I do not know which way is Con-nect-ti-cut!?”
When I was in Kansas with a cousin for three weeks in the mid-70s, in a town called Larned at the junction of the Pawnee River and the Arkansas River, the residents faithfully referred to it as the Ar-kan-sas River, just as spelled. 
I’ve never heard anyone omitting the final “t.”
I’m originally from CT and now live on Long Island. I pronounce it kuh-NET-teh-Kit. I’m starting to pronounce the other one Lawn-Guyland. 
The medial -t- s are odd to me. If I (a Californian) were to spell my pronunciation, it would be Cunneddicut, with the final cut rhyming with put, not cut.
I say “cuh-NET-uh-kit”.
It’s like the teeniest weeniest t in the world, often; if there’s anything behind it, it dissapears, like the -z at the end of my real name (which gets pronounced only when it’s the last word spoken or when the speaker is a foreigner). At least when I was in NH and Philadelphia; I don’t think I ever needed to pronounce Connecticut while in Miami. Then again, I’m still trying to get over how can Americans mute any t at all and I discovered “tweeny” as the pronunciation for 20 in 1987.
And as Gramps Simpson pronounces it:
It’ll be a long time afore I recognises Missora
I think the real question should be: "How do you spell “pronounce”?