Playoff games have gone into overtime 376 times from the 1996-1997 NHL season to the 2015-2016 NHL season (ie, "in the past 20 years")(1).
Using Hockey Reference's Team Game finder, I specified the following criteria: "For single games, from 1996-97 to 2015-16, in the playoffs, in overtime games, sorted by descending Game Date."
Rate of NHL playoffs going to the overtime
In the past 20 years, from the 1996-1997 NHL season to the 2015-2016 NHL season, there have been 1646 playoff games played(2).
Thus, the 20-year average (376/1646) is around 22.8%.
I'd like to see how the rate has fluctuated over many seasons.
Note: results are divided by two to account for each team being represented per game.
It feels like over the last few years ... more games are going to overtime.
Not entirely true. Although there was an higher than average trend from 2012-2014, the average in 2015 and 2016 was behind the 20-year average.
Other seasons of note include 2001 (~5% higher than the next highest percentage outside of 2012-2014) and 2000 (~8% lower than the next lowest percentage and 12% lower than the 20-year average).