This may be all right in the States, but it wouldn't do in Britain. Here we believe in being fully informed if possible, or else well informed, but very informed? Surely not. I suppose the theory behind this convention is that the word "very" is meant to qualify an adjective, not a verb, and you can no more say "very informed" than you can say "very shot at" or "very killed"; but it collapses as soon as you look at it. After all, one can be very depressed or frightened or upset, and these are verbs too. So what is it about informed that makes us reluctant to put a very in front of it?
The answer must be that inform has for most of its career been a pretty muscular verb, by which I mean that it has been about people doing something really purposeful: first, giving form to something, or impressing something on it, or moulding it, then doing the same to people. Teaching them, in fact. In the late Middle Ages, when the word was already being used in this way, teaching and indoctrination were the same thing. The idea that it's wrong to indoctrinate children belongs to the 20th century.
Today the moral implications of the word have quite disappeared (as they have from information) and to inform someone is merely to tell them. But the point is that an informed person is someone who has had something done to them, whereas to be depressed, say, is to be in a state of mind - you don't ask who has done the depressing, you simply think of depressed as an adjective. Of course if you think of informed in the same way, as just an adjective meaning "knowledgeable", then, and only then, you can have your very.