The Braves haven't had an arbitration hearing with one of their players since 2001, which is when they "beat" John Rocker in that case. They came close in 2009 with Jeff Francoeur, and were also on the cusp of heading to arbitration with Craig Kimbrel, Freddie Freeman, and Jason Heyward before all 3 of those players agreed to contracts at the eleventh hour, of sorts.
However, after 14 years, we're probably going to see the Braves head to arbitration with Mike Minor. The difference between Minor's case and the 3 players who received contracts last year is that Minor is probably not in line to receive a multi-year deal right now; Not after the season he had last year, which is when he had a rough year where his ERA ballooned to 4.77 and his FIP also spiked up to 4.39 after having a nice 2013 season where those numbers were at 3.21 and 3.37, respectively.
Of course, injury had something to do with those numbers since Minor had shoulder issues that cropped up as a result of his late start in his prep for 2013 due to having surgery on his urethra (I still get chills down my spine thinking about that. I'm 12 years old.).
Still, injury or not, the Braves probably aren't going to hand him a contract extension after the season he had last year, so that means that they'd rather argue their case in front of the arbitrator. MLB Trade Rumors is projecting that Minor will get $5.1 MM this season, which would obviously be an improvement for Minor over his $3.8 MM salary in 2014.
However, it's no multi-year deal, though Minor could definitely earn that in the future if he bounces back to his 2013 form. But for now, it appears that we're probably going to see the Braves get ready for their first arbitration case in a decade-and-a-half.