When you send a text message through Kennel Connection, the message is sent to the client's carrier's server (Verizon, for example). That's all that KC does. It's up to the carrier to send the message to your clients.
This error means that the message is getting out of KC to the carrier's server but then at some point is being rejected. Common causes are:
Try sending a test text with the word Test as the subject and body and see if goes through. If so, then it's an issue with the subject or content of your message.
Some carriers restrict text message lengths. Click here for details.
If none of this helps, you will need to contact your email provider to get more assistance on why you are being blocked.