6. Given that $S$ is a set composed of $n(n \geqslant 3)$ positive numbers. If there exist three distinct elements in $S$ that can form the three sides of a triangle, then $S$ is called a "triangle number set". Consider the set of consecutive positive integers $\{4,5, \cdots, m\}$, where all its 10-element subsets are triangle number sets. Then the maximum possible value of $m$ is ( ).
(A) 1003
(B) 503
(C) 253
(D) 103
6. C.

Let there be three positive numbers $a_{1}, a_{2}, a_{3} \left(a_{1}<a_{2}<a_{3}\right)$.
If $a_{1}+a_{2} \leqslant a_{3}$, then these three numbers cannot form the sides of a triangle. To make $a_{3}$ the smallest, take $a_{1}+a_{2}=a_{3}$.
$$
\begin{array}{l}
\text { Let } a_{1}=4, a_{2}=5, a_{3}=a_{1}+a_{2}=9, \cdots, \\
a_{i}=a_{i-1}+a_{i-2}, \cdots(i \geqslant 3),
\end{array}
$$

we can obtain a non-triangle number set
$$
A=\left\{a_{i} \mid i=1,2, \cdots\right\} .
$$

Obviously, any subset of $A$ is not a triangle number set. The subset of the smallest 10 elements in $A$ is
$$
A_{0}=\{4,5,9,14,23,37,60,97,157,254\} .
$$

Therefore, $m<254$.
If any element in $A_{0}$ is replaced by any other positive integer from 4 to 253 that is not in $A_{0}$, the resulting new 10-element subset will be a triangle number set.

Thus, the maximum possible value of $m$ is 253.