The "problem children" have only become a big-time issue when what they offered on the field was no longer worth the baggage they brought with them off the field.
Moss was accused of assaulting a woman in 2008, he made remarks before 2010 saying that the Patriots "don't pay" and that 2010 would likely be his last in NE. But he was only shipped out of NE when Belichick felt that his skill set and what he brought to the NE offense was no longer worth the attitude/antics they had to deal with off the field (i.e. those press conferences about how this season would his last as a Patriots).
There were a lot of stories that came out about Adalius being a "problem" from the get-go with the Patriots. He didn't like moving to ILB during 07, he questioned Belichick rather than just do what he was told, he was frustrated early on with his role (playing more coverage than rushing the passer). But Adalius was only discarded of when his level of play didn't equal all the BS Belichick had to deal with off the field.