BL 102-4 receiving calls but not ringing

That's pretty confusing you're not ringing but you're ringing on landline. So, what's not ringing? Have you checked your ringer settings for the cordless phone system? Have you checked Do Not Disturb?

The telephone company guy says our cordless phone is receiving the calls, but it keeps muting it, and he couldn’t find a way to change it.  Our landline does ring, and we can access the call from all the cordless phones.  We can call out on everything and answer on everything, but the cordless simply doesn’t ring. I’ve reset everything, checked the ringer settings, mute, and forward calls.  Nothing is blocked.  Chris, telephone guy, checked our phone from the pole, inside the house and their office.  He tried a new card and even disconnected everything and reconnected it.  

When I say “it keeps muting it,” I really don’t know what is doing that.  Chris determined that the call is being recognized by the cordless, but he could not find a reason as to why it wouldn’t ring.  He thinks the phone might be faulty.

Plug a plane old phone into the jack and see if it rings. What's the service? traditional landline? digital landline? third party voip service? wireless home phone? If this is plane old phone service has it been tested from the NID. If the phone is faulty than plugging in a regular corded phone will help determining that. Did his test set ring when the line was called?

Plain old phone is the landline; traditional landline.  Yes, his test set did ring.

Then if there's no settings you've set to keeping it from ringing, you may have the call blocker set to on and your block settings maybe keeping it from ringing. See page 48 of the manual found here to turn call blocker off, it comes set to on as the default so if you didn't turn it off that could be your problem. Did it ever work?

Unfortunately, nothing has worked.  The call blocker was and is off.  We can answer the phone and dial out; it just won’t ring.  It has never worked for us.

Well, if another phone works when plugged in than you may have a bad base unit. Other than carefully reading the manual on every aspect of the phone that can cause this I have no further suggestions. Some of the causes for no ring can be found on pages 94 and 95

You said you've reset everything; did you check to see if call block was off after resetting? My last question is the caller ID announce working, if you're subscribed to caller ID. 

Yes, the call block is off.  Caller ID announce is not working.  However, the telephone number and name do appear.