Solution:

If $n \le 2007,$ then $S(n) \le 28$ and $S(S(n)) \le 10,$ so $n \ge 2007 - 28 - 10 = 1969.$

Since $n,$ $S(n),$ and $S(S(n))$ all leave the same remainder modulo $9$ and $2007$ is a multiple of $9,$ each must be a multiple of $3.$

Checking the multiples of $3$ between $1969$ and $2007,$ the condition holds for $1977, 1980, 1983,$ and $2001.$

So there are $4$ values of $n.$

Thus, the correct answer is D.