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If you want to delete a whole conversation with one person just swipe right on there name and it'll ask you if you want to delete the conversation. Don't know if thats what your looking to do though.
 
Or you can go to the list of all conversations and tap EDIT and clear each conversation.
 
I believe...

A. You can delete an entire conversation with any person, erasing it entirely.

B. You can delete individual messages in any given existing conversation, one by one.

Other than slowing down or lagging the phone, does anyone know how many text messages can be stored on the iPhone at a time? i.e. my Motorola Razr2 can hold a limited amount <200 in the inbox, but what about the iPhone?
 
I believe...

A. You can delete an entire conversation with any person, erasing it entirely.

B. You can delete individual messages in any given existing conversation, one by one.

Other than slowing down or lagging the phone, does anyone know how many text messages can be stored on the iPhone at a time? i.e. my Motorola Razr2 can hold a limited amount <200 in the inbox, but what about the iPhone?

It's something like 75,000, I read it somewhere around when the 1st Gen was released, if I remember correctly.

EDIT: Found where I read it. It was here: https://www.macrumors.com/iphone/2008/01/18/iphone-1-1-3-hidden-features-bluetooth-issues/
 
I have conversations going from April from when I first got my phone. Will this slow it down?
 
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