Virtually impossible. Williams is going to get about $19-$20 million in the second year of his deal. Joe Johnson will be pulling $21 million per in 2013-14. That's already $40-$41 million, and doesn't count 9 minimum deals for open roster spots under 11. That's going to be another $2 million to $2.5 million. So now we're at $42-$43.5 million. Howard's max deal is about $20 million. Now you're $4 million above this year's salary cap in the best case (includes dumping your first round pick), so your only hope is a huge season in revenue for the league to bring the cap up to $62 million+.
If all their minimum deals were only for a year, they could renounce them next summer to sign Dwight, right?

So let's say Deron gets $19M next year, Joe J gets $21M, and the cap creeps up slightly to $59M. There's still $19M to sign Dwight, which isn't a huge pay cut. Even if it drops to $18M it's still not unthinkable.

Then they'd have have to fill out the rest of their roster with the room exception and a bunch more minimum deals. (I can't remember if they'd get the BAE/LLE as well, so that might be a possibility.)

Signing Joe Johnson doesn't seem like it makes a lot of sense, but it wouldn't necessarily have to nuke the deal for Dwight.