As far as I read about the dev boards, every SoC is capable to use Bluetooth.
I didn't tested it yet, but can I use Android Things with a Bluetooth connection? My question is, how can I enable Bluetooth without an input device? If I want to enable Bluetooth on my phone (with code), I had to confirm it, but this can't be possible on Android Things.
Update: Since the release of Android Things developer preview 3, Bluetooth and BLE are now available.
Old Answer
No. You can not use Bluetooth with the current version of AndroidThings (developer preview 1).
It is said in the known issues part of the release notes that Bluetooth is currently disabled (and so is USB).
It is supposed to be included at some point, but at the moment if you try to get a BluetoothAdapter it does return null.