Can a protein falling below a threshold trigger another response?
Of course, that is possible. I'll give a simple example. If your protein (X) is an inhibitor of the other protein (Y), then when X falls, Y will rise. This would not really be "thresholded". 

There are many mechanisms that can lead to thresholding which include co-operativity (in the action of X) and positive feedbacks. How these mechanisms work would be a different question altogether. (A quick read would be this wikipedia article).