Usually just the opposite for me. I really start to enjoy a cigar around the halfway point. The only reason I would toss a cigar would be because of a hard draw. That will also ruin the taste. Also, never let saliva get into the cigar. As when working in the yard, etc. and stooping over that can easily happen.

Take a La Aurora for example. A medium strength cigar, these really get flavorful around the halfway mark. To toss it before then would be, well, I wouldn't tell anyone that I can't finish a cigar.