My phone does not ring or vibrate with the silent switch.

Does the screen show the incoming call? That is can the phone take calls and answer them, or is it not indicating the incoming call at all?

Is it vibrating when a call comes in?

Have you checked the "Do Not Disturb" settings?

If the phone "sees" the call and displays it on the screen, but doesn't give any sounds my first thoughts relate to volume settings or silent switch malfunction.

When you flip the silent switch (with the phone unlocked and the screen active) do you see the ringer silent, or ringer volume icon? If yes, then it means the switch is working and the problem is elsewhere.

Can you use the phones speaker? Test it by going into settings and test play ring tones, every time you select a tone the phone should play it. If you don't hear anything, recheck the ring volume and DND settings. also see if something is lodged in the headphone jack, this would cause the phone to believe there are headphones installed which would silence the speaker.

Do you have a bluetooth speaker in you car? (like a hands free) They have some significant range and sometimes pair with the phone while you are inside the house. try turning off the BT radio in the phone and try a call.