It's possible but you have to have a ton of help.
Assume the cap will be $101 million.
The current situation is that the Celtics have the following guys with guaranteed contracts:
Horford 27,734,405
Bradley 8,808,989
Crowder 6,796,117
Thomas 6,261,395
Brown 4,956,480
Smart 4,538,020
Rozier 1,988,520
Jackson 650,000 guaranteed
That's 61,083,926 guaranteed
The also have the following guys non-guaranteed:
Zeller 8,000,000
Mickey 1,471,382
Jackson 1,384,750 total
And the following cap holds:
Johnson 15,600,000
Jerebko 9,500,000
Olynyk 7,735,033
Young 2,803,507
Yabusele 2,247,480
Zizic 1,645,200
Green 1,471,382
OK so you cut Zeller, Mickey, Jackson all loose. You have to pay Jackson $650,000, though. And you renounce everyone but Yabusele and Zizic.
That gives you $65,626,606.
Cap hold for the #3 pick is $5,600,000 IIRC, but that only accounts for 8 players. Roster holds up to 12 are $815,615 each.
So now your cap number is 74,489,066.
That only leaves you with $26,510,934 to deal with. If you assume Hayward or Griffin as max players that's $30,300,000 they are looking for. I think it is possible to also renounce the Bi-annual exemption which somehow clears more space, but I'm not an expert. Regardless, even with the $3,290,000 that would add, you're short $500k (and, again, I'm not sure you can really clear extra space renouncing the exemption, and that's a pretty big if). Possible that either of those guys *might* say OK cool we'll take a little hit to play for a contender.
Regardless, let's pretend that it does work this way. Now you're at the cap and you're trying to trade for Butler, which means you have to match salaries. Not with all the guys you renounced...with who is left. Butler is making 18,696,918 next year, he gets a 5% bump for being traded, that's 19.631,763. So the Celtics would have to send $15.750,000 in trade.
So that could be Jackson (4,704,500), Crowder (6,796,117), and Brown (4,956,480)
So now your team is Horford, Hayward, Crowder, Bradley, Thomas...Smart, Rozier, Yabu, Zizic, and vet mins.
I'm not sure this is that smart, even if you can make it work.