Can I start a new conversation with the same number or Apple ID in Messages? Sometimes my conversations get long and unwieldy over time, but being so sentimental I'd like to keep them around on my devices (and not just leave them in iTunes backups that I might not be able to access again later).

In the Messages app, can I start a new conversation with a person under the same number or Apple ID, while preserving our previous conversation on my iOS device? Or if not, is there a feasible workaround (somehow involving switching numbers in the person's contact card or something)?
I think that when you send a new message and it has the same receiver, the messages app automatically groups that with any old messages. The threads are really designed to be conversations between people and not separate threads with new conversation topics like email