A group of us are teaching tying and during the first session found that the threads were breaking with very little tension. Like you, we checked the bobbins and they were not the problem. We realized that the 36 spools of thread we bought (1 box each of 3 colors) were junk. We threw them in the garbage.
Yes, 3/0 is supposed to be double the denier of 6/0. Since you got the thread from a "local dealer" take it back and explain the problem. Oh, I suggest you stay away from the 'kevlar'. Totally unnecessary for probably anything. Might be good for salt water flies but I really can't say. 6/0 is strong enough for almost all applications.

Deezel