Is there a cresent moon by a contacts name in your messages?
If so it means you’ve silenced notifications from that thread of messages. iOS lets your mute individual threads in Messages
Here is how to fix it:

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  1. Click on the message that has the moon next to it.
  2. Click on the “i” icon in the top right hand corner.
  3. Where is says “hide alerts” you can toggle it on or off

When this option is enabled (green), you will NOT get notifications (your device will not vibrate, will not make any sound etc) but you will still receive messages.