According to the Library of Congress, it is possible to download and store text messages on a computer. The method of doing so depends on whether the cell phone is basic or smart.
Basic cell phones must have their SIM card removed before attempting to download the text messages. Once the SIM card is removed, it must be put into a SIM card reader and connected to a PC through a USB cable, where it is possible to download and store the text messages. Smartphones can use a USB cable to connect to a PC without the use of a SIM card reader. It is also possible to use wireless communications, such as Bluetooth, to transfer the text messages.