Yep, FSU is currently riding a 5 bowl game winning streak, the longest active bowl win streak in the nation. They also hold the all-time longest bowl winning streak at 11 straight wins from 1985 - 1995 (UF snapped it with their win in the 1996 Sugar Bowl which also won them a natty).
FSU is also sitting on the longest active bowl appearance streak at 31 (started in 1982), which is the 3rd longest all-time bowl appearance streak. If the streak continues, at the end of the 2017 post-season, FSU will move to #1 on the all-time bowl appearance streak with 36 straight appearances.
And for what it's worth, that 2007 Kentucky team only won by 7, and a lot of that had to do with a shit-load of FSU players being on suspension for the academic cheating scandal that occurred that year.