6. On a plane, there are 2 fixed points $A$ and $B$, and 4 other points $C_{1}, C_{2}, C_{3}, C_{4}$ that do not coincide with $A$ or $B$. If $\sin \angle A C_{i} B - \sin \angle A C_{j} B \left\lvert\, \leqslant \frac{1}{3}(i \neq j, i, j=1,2,3,4)\right.$, then $(C_{i}, C_{j})$ is called a good point pair. Then, such good point pairs ( ).
(A) do not exist
(B) there is at least one
(C) there is at most one
(D) there is exactly one
6. (B).
$$
\begin{array}{l}
\because \angle A C_{i} B \in[0, \pi], \\
\therefore \sin \angle A C_{i} B \in[0,1](i=1,2,3,4) .
\end{array}
$$

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, among $\sin \angle A C_{1} B, \sin \angle A C_{2} B, \sin \angle A C_{3} B$, $\sin \angle A C_{4} B$, at least two values fall into the same subinterval (pigeonhole principle).
$\therefore$ There is at least one good point pair $\left(C_{i}, C_{j}\right)$ that satisfies $\left|\sin \angle A C_{i} B-\sin \angle A C_{j} B\right| \leqslant \frac{1}{3}(i \neq j)$.