Same issue here. I have an iPhone5 that I just upgraded to 6.1.4, and an iPad1 using iOS 5.xx.
Since upgrading to 6.1.4, I can't pair with any of my usual BlueTooth devices (Garmin GPS and JLAB Crusher speaker system).
My iPad has no problem pairing with my JLAB.
Since the iPad isn't a phone, I don't know if it can pair with my GPS, but I'll try later.
My iPhone5 on 6.1.3 had no problem pairing. I've not added any new apps so the only diff is the iOS update and thus the only thing that could be the problem logically is that 6.1.4 has Bluetooth issues.
Swimgrl, did getting a new phone fix your issue? What iOS did it come with? If 6.1.3, then it probably wasn't the phone and Apple replaced yours for no reason. I'm just sayin'
I just trouble shooted with AppleCare and regarding my JLAB Bluetooth Speakers, did the following:
iPhone settings > Buetooth > Devices > click the arrow of any of your non connecting devices.
In the next window > click Forget this Device
Turn your Bluetooth device off and then on and it will show in your iPhone Bluetooth list of discoverable devices
You will notice the type is black and not blue and say "Not Paired"
If you don't see this, turn your iPhone Bluetooth off for 15 seconds and then on again. Then turn your Bluetooth device off and then on again.
Hopefully at this point, you Bluetooth device will show in your iPhone Devices List but you will see the text in solid black. If you have other devices, they may be in blue.
Make sure your bluetooth device is on and then click this device and it "should" pair and say "Connected"
Try your device and it "should' work. Mine does. Now.
My analysis: it seems that upgrating to iOS 6.1.4 throws a curveball into the pairing of bluetooth devices and you have to turn them on and off along with your Bluetooth in your iPhone and then even Forget the Device within iPhone Bluetooth settings. And reinstall it.
Once you do that, if you're like me, it will work just as before. Bluetooth always feels like Voodoo to me, but if you hang in there and keep trying it will eventually work.
Hope that helps.