Yes, you can. In fact, you can make $\tan(N)$ arbitrarily close to your favorite integer.
Pick your favorite integer, which is probably 17. Consider $x=\arctan 17$. Then $\tan(x)=17$ and also $\tan(x+n\pi)=17$ for all integers $n$. If you could make $x+n\pi$ very close to an integer (say $K$), then since the tangent function is continuous, $\tan(K)$ will be very close to 17.
So the problem is now to make $x+n\pi$ very close to an integer. This can be done because the multiples of $\pi$ (or any irrational number) are dense mod 1, so you can make $n\pi$ arbitrarily close to $-x$ mod 1. This takes proof but it is elementary: think about a circle of circumference 1, and start walking around the circle, around and around, marking a point each time your total distance travelled is a multiple of $\pi$. You'll see the marks start to fill in the circle, and as you go on and on, the largest interval without a mark in it keeps getting smaller. Eventually you will mark a point that's within $\epsilon$ of $-x$.