My phone puts a time stamp on the text messages it receives. However, it appears to reflect when the message was sent, rather than what time it actually showed up on my phone, which makes me believe that the time code is sent along with the message through the network rather than being determined by
the phone itself.
I tend to keep my phone turned on most of the time except, of course, in areas where I'm told to turn my phone completely off as opposed to simply silencing it for fear of interference with sensitive equipment (hospitals, airplanes, etc). However, if I'm in an area with no reception or my battery dies, obviously my phone won't receive any messages until it's back in service. Many times in the past, I have left the "wireless black hole" and immediately had my phone vibrate or beep for a text message, which someone sent, say, 30 minutes ago. However, my phone shows the time the message was (presumably) sent, rather than the time that IT received the message.
I could see this becoming a problem if someone thinks I'm ignoring them because I didn't respond for an hour, when in reality, I had to turn my phone off or I was in a basement for a long time. I would have no way of proving that I really didn't receive the message until later.
Every cell phone I've ever had was equipped with some kind of clock. I don't think it would be too hard for the phone to be programmed to read it's own clock as soon as it receives a message, and put that time code above the message so that if one had their phone off or had no signal when the message got sent, they could prove it if their friend/partner/boss questioned them.