I totally agree with you Beth. This is something that all FIRST teams can benefit from.
Our team (1577) and "Hamosad" (1657) have the same teacher that mentors us (he is our only adult mentor, and manages to run both teams by sleeping about 2 hours a night during build season
). The members of our team last year considered them as our biggest rivals, and we had to be better then them at everything (which is a pretty childish attitude). This year the teams really became good friends, and we both help each other out whenever possible (recently they drove an hour with their robot and mini rack to help us out with a demo we held in our city). We also plan on having a barbecue with them when they return from Atlanta.
I think both teams would agree that they benefited greatly from this friendship.