Although Apple has long allowed you to receive calls from specific contact lists when the Do Not Disturb feature is enabled, this applied only to incoming phone calls and required that you designate your exempt contacts as “Favorites” or organize them into specific lists.
With iOS 10, Apple has quietly added a small enhancement allowing you to designate individuals to bypass the Do Not Disturb setting for calls and/or texts on a per-contact basis. The option is somewhat hidden away in each contact screen under the Ringtone and Text Tone settings used to set custom tones and vibrations for calls and messages from each contact. When editing a contact, tap on Ringtone or Text Tone in the contact record; a new Emergency Bypass switch now appears at the very top of each vibration/tone selection screen, which can be toggled on to indicate that sounds and vibrations from calls or messages from that contact should come through even when Do Not Disturb is otherwise enabled.
