They always give a recording when calling back that states the office is closed, even when calling during business hours

Same

401-709-0066 called and left no message, does not match any caller in my phone book. I have gotten several mailings for car warranty extension and believe this is from the same company. I am on the Do Not Call list, or was.

Update:

So I looked up the number and found it to be provided by Windstream Communications. Went to their site and opened a chat to report the scam. I was told "that number is in an area that is handled by a specialized team within Windstream" and was given a phone number to call. That phone number provided is the Paetec Fraud Department Hotline. Windstream acquired Paetec in 2011, so this makes sense. When calling Paetec, I am told the number in question is part of a block that were wholesaled to a 3rd party provider - this is also not surprising. The representative offered to add my number to the do-not-call list, and to forward this information to the wholesaler (who may not care).

TL;DR

I would suggest filing a complaint with the FCC here: http://www.fcc.gov/complaints
and then calling Patec's Fraud Department at 877-999-7705 and selecting option 3

these people keep calling me too - they have my name and the year/make/model of a car I bought a couple years ago. The people I have spoken to will only identify that they are calling from "the vehicle processing department". When asked "department of what?" or flatly "what is the name of the company you work for?" they will not give further details. When asked not to call, they hang up when I am mid-sentence. The last part really confirms they are a scam. This is EXACTLY what the FCC has warned about here: http://www.fcc.gov/guides/auto-warranty-scams
DO NOT GIVE THEM ANY PERSONAL INFORMATION.
File complaints with the FCC.
I'm also considering local legal action.

We get the same thing. Calls from this number and no voice mail or anything. If you call them from a blocked number, the recording says it he office is closed, even though if it is within the listed open times. No company identity is given.

They called about my warranty--asked about my mileage and could barely speak English

We get the same thing.  We are on the Do Not Call list and reported the number to them.  Some people just have too much time on their hands and no brains.

I am getting the same calls day after day. The lady that finally answered and didnt hang up said she was calling about my daughters car warranty. My daughter has sold that car months ago. I am getting tired of these calls. I am also on the do not calls list.. getting irriating.

I doubt it really was Dodge/Chrysler but more likely an extended vehicle warranty scammer trying to pretend to be them.

We keep getting calls from 401-709-0066 - I called it back. A woman answered and I said why do you keep calling and hanging up?   I said this number calls every day and hangs up when I answer.
The woman who answered said she was from Dodge Chrysler and was calling for my husband.. So I said then why dont you say so when I answer the phone??????She said they wanted to know about his 2007 vehicle which is such a joke, they claim they are Chrysler, they cant look up the VIN and see the car was sold years ago - well traded in?  I said that - you are Chrysler and I am surprised you don't know the car was sold years ago.... thank God we changed to FORD - they never bother you with stupid sales calls.

Keep calling me and hanging up. I answer, then "click" is all I hear...who the heck is this?

This number has called me at least twice per day for weeks now. I finally got someone to speak. He identified himself as Rodney Lopez from DirectBuy of Warwick, RI.

I am on the Do Not Call list. I demanded that I be removed from this call list immediately.

I am considering calling the Better Business Bureau.

Calls and after I say hello 2 times, hangs up.