If someone calls you when your phone is fully powered off, the person making the call will hit the cell tower closest to them, but then it will go straight to voicemail with your phone provider. It wont try to ring on your phone because your phone wont be pinging to a tower, so it wont know how to reach you. Thats why it goes immediately to voicemail without ringing.
Similarly, if someone sends you an SMS message, it will deliver the message via the tower closest to the sender. Then if your phone is off, the message will be stored on a server with your phone provider. Then your phone provider will periodically try to send the queue of messages to your phone, and as soon as you turn your phone on and ping your nearest tower, the message(s) will be delivered.
But while your phone is off, there will be no pings. Unless you have some NSA spyware on your phone, then it would be possible to continue connectivity. Or if you have a sleep mode on your phone which resumes network connectivity. When powered off, my phone is completely off with no network connectivity. Same would go for airplane mode; which would cut off all of your network connectivity as well.