Unfortunately, this is not the case with my new iPhone.
Here is my solution:
You can send an SMS Text Message to someone from email. Each cellular phone company has a format to embed the cellphone number into it to create a valid email address. For Verizon that format is xxxxxxxxx@vtext.com where the 10 x's are the cell phone number. Using my cell phone number as an example, if I wanted to send myself an SMS Text Message and a copy of that message to email, I would put in the To field of the email, 4042290839@vtext.com and carlthorne@hthcatlanta.com. The vtext.com address directs the SMS Text Message to the SMS Text Messaging Application on the recipients cell phone and hthcatlanta.com email address sends the SMS Text Message to their email inbox.
Admittedly, this is a clumsy solution and it only captures one side of an SMS Text Message dialogue, but it does allow you to capture important SMS Text Messages without cutting and pasting message elements into an email that I send to yourself.
Here is a link that documents many cell phone provider's formats for sending SMS Text Messages from an email program: http://www.makeuseof.com/tag/email-to-sms/