You can understand the "sub-" prefix as "lower than" or "underneath". So the suggestion of tragedy or difficulty comes from the thing or person being "lower" than what it receives.

A year and a half later, here's my take on it. "Subject [to]" is an adjective, and is commonly used to express a [likely] possibility of something happening. One can often see it in signs: "All passengers entering this area are subject to search" (= "The guards may search anyone entering this area" [but they may not be searching anyone right now, and in fact some people will not be searched]) ; "Road is subject to flooding" (= "This road may be flooded when the snow melts in the spring, or when it rains a lot in fall" [but probably it is not flooded when you are standing here and reading this sign in mid-summer). "Subjected [to]" is indeed the passive form of the verb "to subject [to]". Thus "is/was subjected to search" means that someone is, indeed searched by the guards right now (or was actually searched at some point at the past). "When entering the airport, John was subjected to a search" (= they did search him). "The roots of the plant were subjected to flooding" (= the scientist did flood the roots of the plant to see what will happen to it). So the "subjected to" is more likely to express an actual event -- happening to a particular person or thing at a particular point of time, while "subject to" is more likely to refer to a general possibility (or maybe a habitual. repetitive event). Obviously, actual events are more "dramatic" then potential. Besides, it seems that people mostly use "subjected to" to describe various unpleasant events: "was subjected to search / to humiliation / to torture / to flooding / etc". So that may add to that "tragedy / difficulty" feeling. While "subject to" (as used as an adjective, not as a verb) seems to be slightly more neutral (though there are still some unpleasant connotations).