Do the old OAuth tokens become invalid when new ones are issues?

I am a developer for poptin.com and I am working on an integration with Constant Contact. I am using OAuth to authenticate with your platform. But there seems to be an issue, when the user authorizes a second time, the tokens that were issued on the  first time become invalid. This doesn't happen with OAuth on other platforms. Is this a security thing? In Poptin, the client can choose to connect the Constant Contact integration in a few places, but as soon as they will try to authenticate and connect in multiple places, old integrations will stop working. How to work around this? Is there a way there to make the authentication work with multiple tokens?