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I got an iphone 5 from Telus in September 2013. Over the past year, gradually more and more frequently it has refused to send texts, or has taken a very long time to send texts. This is despite having very good signal. I spoke with a Telus rep who told me to turn LTE off. I have it off but continue to have issues with texts. Also, often times people will text me but I will not receive the text, either at all, or until several minutes later. I once got a text from a friend 20 minutes after he sent it.
I've had texts arrive hours later; especially with Messages. SMS is not an instant message service, though it usually appears that way if network traffic is low enough. It is the nature of the service to accept texts from your phone, store it on their servers until confirmed delivery occurs to the recipient's phone. This can take a varying among of time depending on the situation.
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NFtoBC hit it on the nail again! My only concern is with the "not being able to send texts" bit. Did you try a full reset? Settings --> general --> reset & "reset all settings"? And failing that, a wipe/restore?
If timing is critical, some people resort to third party instant messaging apps, e-mails, although those can have their own issues, and always require data connectivity heh.