Snapchat auto-deletes chats sent one-on-one right after both users open them and leave the Chat. You have up to 24 hours if you change the erase rules in the Chat Settings, but that's it. The servers for the app also auto-delete any unopened Chats after 30 days. However, you can preserve/save them by pressing and holding the Chat. Therefore, you can't read old conversations unless you have already saved them within the allotted time.