The short answer is no (this has been beaten to death elsewhere, if you want to look around, thanks to Gary Tanguay's incorrect claims that it could be done).
Even if Pierce did opt out, the C's would still have something like $42 million on the cap eaten up by Sheed, KG, Perk, Rondo, and Davis...and that doesn't even get into the cap holds, etc.
The highest I have seen the cap projected at next year is $53 million. So that leaves them $11 million (minus cap holds), which would not be enough to sign Lebron...and even if they did convince Lebron to take less, they would have nothing left to resign Pierce for anything more than the vet minimum (and would you give up $20 million?), since teams under the cap do not get to use the MLE.
It is a fun thought, but alas, it is not possible.