If even resetting your Network Settings couldn’t yield positive results, it’s time to give your carrier a call or visit them.
On the iPhone, go into voicemail and tap "Greeting". Record a few seconds of something and tap stop. Before you tap save, use iFuntastic or SFTP to navigate to /var/root/Library/Voicemail on your iPhone. You'll see a file called "Greeting. amr". This is the audio you just recorded. Delete that file and replace it with your custom message which should then be renamed to "Greeting.amr". Tap "Play" on the iPhone to verify your message works. If it plays, tap "Save" and your custom message will be uploaded to AT&T's servers.

Android phones can access voicemail by holding down the 1 key on the dial pad until the phone dials your 10-digit phone number. You will automatically connect to your mailbox and be prompted to enter your temporary password; this password is the last 4 digits of your phone number followed by the # key. Start the Google Voice app. Tap the three horizontal lines at the top left of the screen (sometimes called a hamburger menu), and then tap “Settings.” In the Voicemail section, tap “Voicemail greeting.” Tap “Record a greeting.” How do I access my voicemail? To call your voicemail on an Android phone, simply open your phone’s dial pad and hold your finger down on the “1” key. You can also call your voicemail from a different phone by calling your own number and tapping the pound key. How do you check your voicemail?
Hello, you have reached the reception of Dr. X please leave your name, reason for calling, and your number so I can call you back as soon as I can. If you are experiencing a medical emergency please hang up and dial 911.