iPhone Won't Ring, Delayed Voicemail Indication

Greetings.

I have replaced my iPhone once already. The last one had a strip of frozen pixels across the touchscreen. I have had this new iPhone for a few weeks now. I would guess that out of, say, 8 calls I actually get a ring on one of them? Most of the time I pick up the phone to check something else and get the "Missed Call" indicator. If the caller left me a voicemail it will not indicate for hours, sometimes days. If I turn the phone off and on I will get the voicemail to indicate once the phone has powered up. From a forum search I gather having my phone set to autocheck for e-mail every 15 minutes could be one reason I am missing calls. It seems a bit improbable that every call that comes through happens to be at the exact moment the phone is accessing Edge though. The voicemail issue baffles me.

The previous iPhone I had was set to check e-mail every 15 minutes and I did not have this problem. I got a ring on every call and voicemail indication was instant. My instinct is to call AT&T but I fear this may be an issue with the phone. Of course if I take it into the store everything will work perfectly.

Any ideas? Should I turn off e-mail autocheck?

Thanks.