No, Snaps recovery isn't available unless you have actively tapped on each message to turn it to grey color. This is practically saving, although you can't save the entire conversation.
Snapchat was build in such way to protect privacy to and make the conversations more private. Once a message has been read by both parties, Snapchat will delete them after you focus on another chat.
You might want to try Viber or WhatsApp. Those applications can backup and restore conversions.