My story is a bit different, but possibly helpful in understanding the issue. I had a T-Mobile HTC One S running ICS. I got a 2013 Volt with the full Nav package which includes Bluetooth. In the case of GM they recently turned over the support of Bluetooth to On-Star. GM stopped doing the software builds themselves for all GM brands. Long story, short... I had the same connect/disconnect issues + garbled communication problems many of you are having with your GS3's on jelly bean. I worked very closely with the special GM Volt customer service team on the issue. What they determined is that On-Star was using older Bluetooth protocols that worked only with certain carriers versions of ICS. For example, it was comparable with the AT&T HTC One X, but not the Bluetooth software on the T-Mobile One S. On-Star then updated the Bluetooth software via an OTA update and my One S began to work perfectly. Clearly it was an issue with the software On-Star was using.
Fast forward two months and I now have upgraded to a GS3 that came pre-loaded with 4.1.1. I am having absolutely no issues with the Bluetooth program in the Volt. In fact I have a TecTile setup in the car and it is set to turn Bluetooth on and off in the car. Again, no issues with pairing and loading my contacts into the system on the car. My feeling is, based on my very frustrating experience with the One S and On-Star, the issues lie with the automobile manufacturers, or whom ever builds their Bluetooth software for their cars, and not the phone manufacturers or carriers. It is the automobile manufacturer's responsibility to insure the Bluetooth protocols they are using are comparable with the various OS builds...not the other way around.