If everything had broken right for the Celtics (i.e., Rasheed retired with no money left on the cap), the Celtics would have been looking at approximately $16.5 million in cap room, and would have had to renounce all their free agents.
Pierce's new deal is, at minimum, over $13 million in the first year. That swallows up whatever cap room there would have been, meaning that the only outside free agents we sign will be with the MLE or the minimum salary.
(We can also upgrade through trades and sign-and-trades.)