Yes Boston could but the trade for Cousins would have to include near equal salary of players (or picks) on the books for next year. Cousins makes 18.1 next season so that is the salary you would have to come up with. The 2017 1st rounder would likely be in the trade and that salary could be pretty high. To get the talent level enough you probably need to include Smart and Brown and a future 1st (maybe Boston's 2018). So let's take a look at the salaries if that is the basic trade.
So next year's salary numbers (with Cousins)
Horford 27.7 million
Cousins 18.1
Crowder 6.8
Thomas 6.3
Rozier 2.0
Jackson 1.4
Mickey 1.3
Zizic & Yabo 2.5
Cap Holds ~2
Total = 68.1 million (and Jackson, Mickey, Zizic, Yabu could all be let go but you would need basic cap holds for their spot)
So that means there would be enough space for Hayward, but no KO, Amir, Zeller, JJ, Green as they would all have to be waived to create the room. It would definitely be worth while as that would give Boston 5 all star level players in the starting lineup with Rozier and Crowder on the bench. Team is very thin pending other free agent moves, but maybe a guy like Johnson doesn't want to leave and re-signs for the vet minimum and other moves of that ilk giving Boston enough depth, but Thomas, Bradley, Hayward, Horford, and Cousins would be a scary starting five for the rest of the league to deal with.