Among $\{1, 2, 3, 4\}$ only $3$ works. Call an integer good if it satisfies the desired property and also it is $\not\equiv 3\ (\textrm{mod}\ 4)$. Let $k \ge 5$ be the smallest good integer. Then $k = a + b + 1$, with $a, b \equiv 3\ (\textrm{mod}\ 4)$, but then $k \equiv 3\ (\textrm{mod}\ 4)$, contradiction. So there are no good integers. $2016$ is divisible by $4$ and hence doesn't work.

Suppose we can meet the condition. Then the number of cards left is divisible by $3$, and since we put one card into the hat in each move and $3\mid 2016$, the number of moves made is divisible by $3$. Let $3k$ be the number of moves. Then the number of piles left is $3k+1$, and the number of cards left is $9k+3$ (!). Since we made $3k$ moves, the number of cards in the hat is $3k$, and the number of cards left is $2016-3k$ (*). Combining (!) and (*) we get $9k+3=2016-3k\Longleftrightarrow12k=2013$, which is not true for $k\in \mathbb{N}$, therefore we cannot meet the condition.

Let the number of piles be $x$ and the total number of cards not in the hat be $y$ . On each turn the number of piles increases by 1 and the number of cards not in the hat decreases by 1. Therefore $x + y$ doesn't change and will always remain at $2017$. Also, we are trying to find solutions for $y/x= 3$ . This means that $y=3x$ and therefore $x + y = x + 3x = 4x = 2017$. We see that this is not possible so far as $x$ is an integer, so it is not possible.