I think it's also worth noting that some firms are, simply put, incredibly rude with no desire to behave professionally towards people they don't happen to want.
Of all my interviews and bids--from ones that led to callbacks to ones that did not--only one particular firm never sent me any kind of notification whatsoever after my screening interview. It was a place I was pretty interested in, but the fact they've done this to a lot of people (it seems) who didn't get past the screen has caused their stock to plummet in my eyes, even if it doesn't happen to matter for me personally without a callback.
Yeah. Some of these places are really, really, really rude. So it could be that.
But if they gave you an estimate--"it might not be until well into September"--and you don't happen to know of people who have gotten callback offers from them already, perhaps this is just the time it takes this particular firm and it wouldn't hurt to check in.