The funny thing is that most of the close calls (if I am remembering correctly) happened over 20 years ago, though I would assert that the overall talent gap between top and bottom has diminished to an extent.
This is an interesting question and I did some more checking.
http://en.wikipedia.org/wiki/NCAA_Men's_Division_I_Basketball_Championship_upsets#15_seeds
In 2011, the tournament expanded to the first four game 0th round (which is the terminology that they should use).
Since then, the pool of 16 seeds has been diluted with the worst 4 teams whittled down to 2 in those games. 2011-2014 saw 16 first round 1/16, 2/15, 3/14 games. I got my numbers below from counting in the above link and chose two timeframes - 2011-2014 (4 years with the play in game as we have it now) and 2007-2010 (the same length of time before it was changed).
No 16 seeds won.
3 15 seeds won - ~19% success rate. Compare that to 0 winds in the prior 4 years (and 0 winds since that fateful loss that shall not be named)
2 14 seeds won - 12.5% success rate - also better than the 1 win in the prior 4 years.
3 13 seeds won - ~19% success rate - 4 of them won in the prior 4 years.
9 12 seeds won - a 56% success rate - only 6 won in the prior 4 years. I think this is the area where the play in games are showing more effect because they are adding more average type teams or underperforming teams to the tournament.
8 11 seeds won - a 50% success rate - only 6 won in the prior 4 years.