One of the problems with Cathy is that she seems to be so unlikable, in spite of the fact that she is such a strong personality. The evidence for this is ample: consider the way she insists on remaining married to Edgar but also having Heathcliff as her special friend. Consider how she talks in Chapter 12 after the confrontation between Heathcliff and Edgar, and how trivially she talks of ending her life to spite her husband:
"If I were only sure it would kill him," she interrupted. "I'd kill myself directly! These three awful nights, I've never closed my lids--and oh, I've been tormented!
One the one hand, while the reader might feel Cathy is a character who has a certain romance about her because of the strength of her emotions, if her relationships with both Edgar and Heathcliff are examined, we can see she is very selfish and manipulative, determined to sacrifice anything and everything to get her own way. Remember she chooses to effectively starve herself to her own death out of spite at Edgar. This makes her a character who is not tragic but also neither is she truly pathetic.