Hi @mitchbuchanon! I see you’re working with my good friend - the Twitter API!
What you’re looking to do should absolutely be possible within Postman! In order to do this, you’ll have to have some custom JavaScript code that continuously calls for older tweets if they’re available. This can be done by capturing the “next token” of a request and using that for ensuing requests.
I wrote in more detail about how to achieve this in this post here. Hope that helps!