Every team regardless of cap situation gets exceptions. There are exceptions for luxury teams (how GS got Cousins), over the cap teams (how NOLA got Randle), and under the cap teams (how Lakers got Beasley or Lance I'm pretty sure).
You'll see it a good bit this summer. Let's just say the Lakers use their max space to sign KD. And let's say that puts them right near the $109 salary cap. The NBA's system allows for a team in that situation to continue to build the roster with contracts other than just minimum deals. They wouldn't make the Lakers settle for tons of minimum deals just like how there was a bs loophole for well over the luxury GS to have $5+ to give to Cousins.
So how does that apply to us? In our wildest dreams, we could renounce Rubio, decline Favors and sign Tobias to the max. After we do that, we are at the cap but aren't technically an over the cap team because we started the process having cap space. We could still then sign a player using the "Room Exception" of $4.7 or the "Bi-Annual Exception" of $3.6.