Running faster than your buddy does not a team make

Our most recent project at work was kicked off not primarily with a clear mission and requirements but instead with a date.  It wasn't this blatant: "I want X by Y and neither X nor Y are negotiable." but it was pretty close.  So rather than wasting another second we started sprinting to figure out how to get there.  The "bold date" strategy seems to have worked in the past.  In our case the date seems a little too bold.  No one I've talked to thinks that it's an achievable date.
The greater tragedy beyond achieving or not achieving the goal is that it is creating unhealthy competition.

When hiking with a friend if you encounter a bear you don't have to run faster than the bear you just have to run faster than your friend.

We look at solutions and here's how the thinking goes (whether it is stated or not).

1. What all do I have to do?
2. Can I finish it before Y?
   • Yes:  Yow...  Poor team Z, no way they can possibly hit Y.
   • No: Uh oh...  How can I shed work on team Z or otherwise cut corners?

I'm not sure if this dynamic is the result of a cultural pathogen that needs to be cured or if it's because everyone sitting around the table is thinking "there's no way!".  Makes me really curious what all of the engineers, project managers, accountants, etc. were thinking when Kennedy set such a bold goal.  Also makes me wonder how bold it was.