@Ilya: well, a subset $v$ of vertices such that |$v$|=$n$ (the 6 lines in your example) is said to be shattered if you can get every possible subset of $v$ by intersecting $v$ with edges of the hypergraph. meaning you have $2^n$ distinct intersections. in your 6 lines example, i cant think of a hyperedge interesecting with the 6 lines that would result in the 3 lines not consisting of the small triangle, so this is not a shattered group, and it doesnt show that the VC dimension might be 6.

What makes 10 an obvious upper bound?

Those six lines are a good example of a groupthat cannot be shattered. But in order to prove that the VC dimension is indeed bounded by 6, It is neccesary to show that every group of size 6 cannot be shattered.