To shift political gears somewhat, our legislature is currently engaged in a debate about tax reform. One party claims the currently proposed plan will cut taxes on the middle class. The other party claims that the plan will raise taxes on the middle class. That's a question of fact, and both positions can't simultaneously be correct. The thought process you engaged in above can be applied equally well to that situation regardless of which party one favors, but it's guaranteed to make at least one party's fans wrong.
When we, as a polity, begin to completely disregard information because it comes from a partisan source or has strong partisan implications then I worry that we lose the idea of a shared reality. We can't convince those who disagree with us, nor allow ourselves to be convinced, if we refuse on general principle to listen to and engage with people who disagree with us.