How old was the oldest voicemail? Ever since getting my 1st cell phone circa 2000, there's always little hiccups when it comes to timely/immediate delivery of voice mails and text messages. Usually just an hour or two and occasionally in excess of 24 hours. I don't think this is a problem that is specific to a particular carrier, phone manufacturer, or model. There's hundreds of millions, if not billions of people using cell phones around the world. That's a lot of data being transmitted through space and just like our road and highway system, sometimes there's a lot of back up and it takes you longer to get from A to B.
I have found that if I'm ever in an area without signal for more than an hour or 2 (I live in the sticks, so this happens often), there will be significant delays in messages being pushed through to my phone once I reestablish signal. But if I reboot my phone after getting signal back, my messages are delivered almost immediately upon restart.