6. On the plane, there are two fixed points $A$ and $B$, and 4 moving points $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$ that do not coincide with $A$ and $B$. If $|\sin \angle A C B - \sin \angle A C, B| \leqslant \frac{1}{3}, (i \neq j, i, j=1,2,3,4)$, then $(C_{i}, C_{j})$ is called a good point pair. Therefore, the number of such good point pairs ( ).
A. does not exist
B. has at least 1
C. has at most 1
D. has exactly 1
6. B

Since $\angle A C_{i} B \in[0, \pi]$, then $\sin \angle A C_{i} B \in[0,1]. (i=1,2,3,4)$
Divide the interval $[0,1]$ into three segments: $\left[0, \frac{1}{3}\right],\left(\frac{1}{3}, \frac{2}{3}\right],\left(\frac{2}{3}, 1\right]$. Then, $\sin \angle A C_{1} B, \sin \angle A C_{2} B, \sin \angle A C_{3} B, \sin \angle A C_{4} B$, at least two values fall into the same subinterval (pigeonhole principle).

Therefore, there is at least one pair of good points $\left(C_{1}, C_{3}\right)$ that satisfies $\left|\sin \angle A C_{i} B-\sin \angle A C_{j} B\right| \leqslant \frac{1}{3},(i \neq j)$.