I too have this issue and it's very frustrating. I don't think Apple could have made this any more confusing and poorly designed (which I'm finding out is pretty common for most things that they do). Above it was mentioned that the silence is only an issue when the Do Not Disturb is turned on, but that's not true. I don't have the DND turned on and I still am missing calls, which with much research I'm finding out is when the phone is locked.
When you go to the DND screen in settings the next very frustrating thing is that the only options you have are 1) Always or 2) Only while iPhone is locked. Seeiing this surely also raised the blood pressure even higher as in usual iPhone fashion they don't provide me with the option I want, which is to never silence the phone. Or do as BB does and give me different profiles to choose from (i.e. Loud, Normal, Silence, Vibrate etc...).
So needless to say, I'm now experimenting on how to have this setting bypassed. I'm guessing that you need to select Allow Calls From: Everyone. I don't know. No one here seems to have a good answer for this, so I'm trying to help others as I struggle with my own iPhone frustrations. I hope this helps. I've lost count of how many reasons I'm frustrated daily with my iPhone and the way Apple does things. If anyone has any success with this please let me know.