I'm not entirely sure that the narrator was indeed as careful and cautious as he thinks he is. In the first paragraph of the story, the narrator admits that his sanity has been called into question.
". . .very dreadfully nervous I had been and am; but why will you say that I am mad? The disease had sharpened my senses --not destroyed --not dulled them. Above all was the sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell. How, then, am I mad? Hearken! and observe how healthily --how calmly I can tell you the whole story."
Right away in the opening paragraph you have a narrator that is telling his reader that he hears things from heaven and hell. The entire sequence reads like he is trying to prove he is not crazy. It's like he is begging the reader to believe him.
So when he repeatedly tells the reader how careful he was, I doubt certain parts of it. But in his mind, he was exceptionally cautious.
"Then, carefully, I lifted the cloth, just a little, so that a single, thin, small light fell across that eye. "
The narrator describes how he sneaked into his master's room for SEVEN nights in a row and carefully uncovered the eye. The reader is meant to believe that the repetition of the event shows how careful and patient he was. On the 8th night, he was even more careful. The narrator says that the hands on the clock moved slower than him.
"The eighth night I was more than usually careful as I opened the door. The hands of a clock move more quickly than did my hand."
After the narrator kills the old man, he again stresses that he is NOT crazy. And again, the narrator uses his caution to justify that he is not crazy.
"So I am mad, you say? You should have seen how careful I was to put the body where no one could find it. "
That paragraph alone uses the word "careful" or "carefully" four times. He was careful to not let any blood get on the floor. Careful to pull the floor boards up and careful to put them back. Careful that nobody could see what was done.
In the narrator's opinion he was cautious because he was patient, and because he was careful to hide all evidence of the murder. The only thing he wasn't careful about was his own guilt ratting him out to the police officers.