A chain of edges can close iff the longest edge is not longer than the sum of the lengths of all the other edges. This is Theorem 8.6.3 (p.326) in Computational Geometry in C and Theorem 5.1.2 (p.61) in Geometric Folding Algorithms. You can easily see the necessity of this condition: If one edge $e$ is too long, the others all together cannot reach from end-to-end of $e$.
This result has nothing to do with $6$; it holds for $n \ge 3$ edges.