We are looking for the numbers $n$ such that $200 < n < 600$ and $n$ is divisible by 4, 5, and 6.
This means that $n$ must be a multiple of the least common multiple of 4, 5, and 6.
We find the least common multiple (LCM) of 4, 5, and 6.
The prime factorization of 4 is $2^2$.
The prime factorization of 5 is 5.
The prime factorization of 6 is $2 \times 3$.
The LCM of 4, 5, and 6 is $2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$.
Thus, $n$ must be a multiple of 60.
We want to find the number of multiples of 60 between 200 and 600.
The smallest multiple of 60 greater than 200 is $60 \times 4 = 240$.
The largest multiple of 60 less than 600 is $60 \times 9 = 540$.
The multiples of 60 between 200 and 600 are:
$240, 300, 360, 420, 480, 540$.
These are $4 \times 60, 5 \times 60, 6 \times 60, 7 \times 60, 8 \times 60, 9 \times 60$.
The number of multiples is $9 - 4 + 1 = 6$.
Therefore, there are 6 numbers between 200 and 600 that are divisible by 4, 5, and 6.
The multiples of 60 are:
240, 300, 360, 420, 480, 540
There are 6 multiples of 60 between 200 and 600.
Final Answer: The final answer is $\boxed{6}$