Defining 'swing/battleground' as any state that receives significant attention from both candidates. In 2012, there were 10 states fitting such description: OH, VA, FL, PA, NH, IA, CO, WI, NV, NC
In 2016, there are different swing/battleground states depending on the matchup, but the number is generally greater than 10, so yes, there will likely be more battlegrounds than there were in 2012. A few examples:
Clinton/Christie - MT, NV, AZ, CO, MN, WI, IA, AR, OH, PA, NJ, NH, VA, NC, GA, FL (16 states)
Clinton/Walker - MT, AZ, CO, WI, IA, AR, OH, PA, NH, VA, NC, GA, FL (13 states
Biden/Bush - All 2012 plus OR (11 states)
Biden/Santorum - AZ, CO, MT, IA, OH, PA, VA, NC, GA, FL (10 states)
I want to restate the definition of "swing state".
Swing state is a state that can swing the election. For example, North Carolina was
NOT a swing state in 2008 or 2012 because Obama would have already won the election if the won North Carolina.