I have multiple phones on my account, and all of them frequently can't send text messages. Failures are most common when sending between these phones. Sometime one message will be stuck "Sending", and a new message to the same recipient, with the phone in the same physical location (and 3+ bars of service) will go through.
Tapping on "Not sent. Tap to try again." never works.
Best answer by tmo_mike_cView original