Patrick Mooney has a piece up (link below) noting that Cubs tried to trade for Papelbon before trade deadline but were outbid.
Big difference between Cubs and Nats is that Cubs have guys (Ross, Lester, Maddon) who maybe could have reined in Papelbon's worst tendencies and Nats did not. A guy look Ross might well know beforehand that Papelbon was mad at a teammate (as Papelbon was at Harper following Harper's remarks about the HBP) and headed the whole thing off beforehand.
Not really a clubhouse chemistry thing but a matter of having team leaders and a manager who knows what's going on in the clubhouse.
Highly recommend the 3-part series in Washington Post about the Nats. Writer was working on the series all season and had permission to publish when Nats mathematically eliminated. Good stuff in all 3 parts about roster construction, clubhouse issues, and grind of 162 game season.
Mooney piece:
http://www.csnchicago.com/cubs/what-if-cubs-had-traded-jonathan-papelbon