The Falcons have not made a habit of trading down over the last several years, and despite the obvious incentive to do so in a year where they only have five picks, we don't expect that to change in 2016.
In all of Thomas Dimitroff's years at the helm for the Falcons, they have only twice traded down (silly Dave!). There are teams who prioritize doing so to pick up more selections, but Dimitroff seems to believe (for better or worse), that you're better off going up and getting the man you want. That didn't change a year ago, when the Falcons traded up to grab Grady Jarrett, which looks like a great move at the moment, and it would seem to be a drastic change in the team's philosophy, even if Dimitroff's power in the building seems to be weakening.
The only reason to believe otherwise, again, is the limited number of selections. If Atlanta has an interest in a safety like Keanu Neal and a second round pass rusher or linebacker, swapping down to the late first might be the best way to land both.
Maybe we'll be pleasantly surprised, because Atlanta's odds of landing multiple quality players goes up with more picks, even in a weak draft class. Just don't expect it.