Does it strike nobody else that the OP's question is very odd? The ratio of people who get on the mat with overly stiff arms to those who get on the mat with overly limp arms is, it seems to me, about 1000:0.
It did strike me as uncommon, but I've trained with a person who I would describe as almost completely limp. No amount of asking to try with more force would yield more than the slightest sense of contact even. I've felt it when people just grab on lightly and don't do anything; what I'm remembering was somehow even less than that.
I'd like to hear more from the OP as well though.