My iPhone 5 has a strange problem with voicemail. If someone places a call to me, it will go right into voicemail, when it shouldn't (I'm not on the phone, and my phone is not dead). If they call me right back, the call will go through. I.e., initial calls go into voicemail; a second call from the same number will go through. So if the person knows about the problem and calls me right back, I can get the call. Otherwise, I just get messages. Why would voicemail pick up on a 1st call, and then not pick up on a 2nd call from the same number (provided the 2nd call is made right after the first)? I have it set to pick up after 4 rings. Weird issue.
Make sure you do not have "Do Not Disturb" turned on. This feature will block the first call, but is someone calls right back it will be put through.
On the iPhone, go to Settings/Do Not Disturb
Make sure you do not have Do Not Disturb turned on manual or scheduled.