Why Your iPhone Won’t Play Voicemails — The Fix! Close And Reopen The Phone App. As I mentioned earlier, one of the common reasons why an iPhone won’t play voicemails is because the Phone app isn’t working properly. Closing and reopening the Phone app allows it to “shut down” and start again, which can sometimes fix a software glitch.
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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.
However, some cellular service providers allow users to access and update their voicemail by calling their own numbers from working landlines or phones on other wireless networks, so users whose accounts include such an option and who have access to alternative service could keep the voicemail greetings updated even when their own mobile phones cannot connect to cellular or internet service.