Last September, a new thread in Apple Support Communities briefly noted that a user was unable to get his iPhone 5 to pair with other Bluetooth devices. As it turned out, he was not alone. Cut to the present: this thread is now 29 pages long, with several hundred confirmations of the symptom.
Here's the problem: In the most common scenario, users are unable to pair an iPhone 5 to a Bluetooth car audio system. In some other cases, users are unable to pair an iPhone 5 to a Bluetooth headset. Pairings to other devices may or may not work. Typically, problems begin after upgrading from an older iPhone model (which was working just fine) to an iPhone 5, running iOS 6. In a few instances, the same sort of pairing problems have been reported for the iPhone 4S.
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If this has happened to you, one or more of the following fixes should get you connected again:
Update the car audio firmware: Several postings in the above Communities thread cite confirmations from automobile manufacturers that their cars' audio systems are "not compatible with iPhone 5." In particular, the older Bluetooth drivers in these systems are not compatible with the Bluetooth 4.0 standard used by the latest iPhones. A firmware update to the audio system eliminates the problem. Check with the manufacturer of your car or audio system for details.
Apple has confirmed that some car audio systems require a firmware update to work with AVRCP 1.4 Bluetooth. However, the symptoms described in the Apple article refer only to an inability to perform certain music playing functions--not an inability to pair devices. So I doubt this is a cause of the problems described in the Communities thread.
Follow the instructions: Several users were able to pair their new iPhone to their car's audio system only after they made sure to precisely follow instructions. In particular, pairing required a confirming action from both the iPhone and the car audio system. Users, apparently having forgotten how they originally paired their older iPhone model, were erroneously attempting to pair their new iPhone 5 by confirming only on one device. Similarly, when attempting to pair an iPhone to a Bluetooth headset, some users didn't realize that it was necessary to hold down the headset's call button.
In other words, RTFM. If you can no longer locate the manual for your device, check the thread cited above. Numerous posts offer step-by-step instructions for how to pair different devices to an iPhone.
Modify iPhone Settings: A number of users had success by making a change in the iPhone's Settings app. Specifically, navigate to General > Accessibility > Physical & Motor. From here, change the Incoming Calls setting from Default to Headset. This change ensures that the iPhone creates a Bluetooth headset pairing to your car's audio system, which is needed for phone calls to work as expected. When you are not in your car, the iPhone reverts to its "normal" behavior.
Clear the device's memory: A few users claimed that, when trying to pair an iPhone 5 to their car's audio system, the system would instead continue to attempt to connect with a previously paired, now absent, iPhone model. To fix this, wipe the car system's memory clean, having it "forget" all previously paired devices. After doing so, pairing with the new iPhone should succeed.
Update to iOS 6.1: In some cases, the culprit may be a bug in the iOS 6.0 software. At least for Honda CR-V owners, upgrading to iOS 6.1 seemed to eliminate Bluetooth pairing problems.
Consider the hardware: As it happens, my wife's iPhone 4 fell victim to an intermittent Bluetooth failure. About half the time, the phone would successfully pair with the aftermarket Sony Bluetooth audio system in her car. At the other times, the iPhone would fail to pair. None of the above suggested solutions had any effect. What did work was dumping the iPhone 4 and moving up to an iPhone 4S. In this case, I suspect there was a Bluetooth hardware failure in the iPhone 4.