The opening lines of "The Black Cat" by Edgar Allan Poe are spoken by the narrator of the story, and he insists he is not insane; however, there is sufficient evidence to make the case that the narrator is suffering from some kind of mental illness.
He writes the story from jail, and this is how he begins:
For the most wild, yet most homely narrative which I am about to pen, I neither expect nor solicit belief. Mad indeed would I be to expect it, in a case where my very senses reject their own evidence. Yet, mad am I not -- and very surely do I not dream. But to-morrow I die, and to-day I would unburthen my soul.
This is the night before the narrator is going to lose his life as punishment for the crime of killing his wife, and he assures us he is only telling the "homely" story because he wants to unburden his soul. There are two motives for such an unburdening, and it determines whether his account is truthful. A condemned and guilty man (which is what the narrator is) may wish to write a confession of his actions, without any justification or attempt to mitigate his actions; or such a man may wish to write his version of the story in an attempt to garner sympathy from his audience and excuse his own behavior.
This narrator tries to excuse his actions by claiming that "from my infancy I was noted for the docility and humanity of my disposition." This obvious exaggeration is clearly an attempt to make himself the victim in his wife's murder. He tries to blame his actions on a cat--two cats, in fact--and on his wife; yet he demonstrates a blatant tendency to intemperance (impatience) and overreaction after several years of contentment with his wife and Pluto.
The truth is that he underwent some kind of change which precipitated Pluto's subsequent change in temperament. The narrator's change came first:
I grew, day by day, more moody, more irritable, more regardless of the feelings of others. I suffered myself to use intemperate language to my wife. At length, I even offered her personal violence. My pets, of course, were made to feel the change in my disposition. I not only neglected, but ill-used them.
His mental state is questionable here, and the symptoms are indicative of some kind of mental illness. From this point, the narrator's actions are all affected by that illness. Is he insane? Perhaps, but he certainly suffers from some kind of mental illness.