launch day n6 from goog. stock not rooted.
paired to a moto 360, laptop, mac desktop, nexus player, vehicle OBD II port Bluetooth dongle (elm 327) pop Bluetooth headset, htc bt headset, sony srs-x9, vehicle head-unit, nexus 9 lte, Bluetooth keyboard and zero issues with re-connecting to these when the peripheral requests the phone.
i dont have all of these as trusted devices but the ones that are keep the phone unlocked per expected.
maybe un pair your devices, and re-pair? perhaps a factory reset?
are you rooted?
does not sound like a physical issue with the phone, could just be the devices are not connecting properly, especially if you are able to pair the devices initially.
could be a rogue program causing you issues.
if you are within 14 day grace period, you should be able to get a new device. if past that, you will have to exercise your insurance (you DO have insurance!?) or warranty with moto directly.