I don't know if there is a definite 'tipping point' for AJ, but he is under a bit of pressure. Everyone recognises that we should be doing a lot better than we are given our (suspected) budget and with our squad which, on paper, is better than we've shown. This week's game at Dumbarton is a big one, however, and the fans will lose patience. If we win then things could look a bit more promising and the pressure will ease on him.
As said above, it might be the case that we just aren't in a position to be sacking and hiring management staff.