Not that I would consider it to change my answer, but the term must first be universally defined; the word "sympathy" is often, albeit incorrectly, interchanged with "empathy":
"I have great sensitivity to your concerns."
for instance. The term is also colloquially misused as someone who is particularly (often to a point of fault) reactionary to emotional stimuli. Given the description you gave, I get the impression you are using the term in a tangential reference to the latter; however, you also seem to be using the term to connote an instinctive inclination towards acquiescence in confrontations. None of these are appropriate definitions for the term "sympathy", really.
Either way, it also heavily depends on what you mean by "is it good". It would highly depend on what you value in your life. There are certainly situations in which it's advantageous (or, to be blunt, a facsimile of such is), but typically it is not. To operate on a purely logical plane, as opposed to one interrupted by emotion, is, however, an abhorrent thought to most: "callous", "icy", "unfeeling" etc. In a vacuum, speaking from a purely animal point of view, sympathy is entirely useless; only the feigning of such could be advantageous. However, people do not live in vacuums, nor are they dogs or monkeys. An example like this also does not take into account that other peoples' sympathy is advantageous to you, but getting into things like that is a bit pointless on a place like this. To be too "sensitive" is, of course, negative, but to be entirely devoid of sympathy (as you define it) would define you as a sociopath, which is probably not something you're interested in being. It's really up to you to decide what is "too" sensitive. As I said, no Darwinian model will tell you that genuine sympathy is useful, so it really depends on how monstrous you're willing to be, assuming you have a choice in the matter.
Topic: Is sensitivity really such a good thing?