My husband and I have two cats, both part of our family before the little one came along - one cat is horribly shy (I am convinced she has a kitty anxiety disorder) and pretty much lives in my closet and only comes out once the kiddo is in bed at night. No worries there. Our other cat is large, independent, VERY friendly and affectionate, but is sometimes unpredictable in her mood. She also has a rather quick temper, this cat, so we certainly had our worries about the impending toddlerhood of our daughter - suffice to say she is now 27 months old, and she and our black and white cat are really great friends!
I grew up with cats, but never remembered how I "learned" not to cross the line - my mother let me know that a run in with teeth and nails vs toddler is inevitable, but it usually just takes one incident for the toddler to learn what the cats limits are. One evening when my daughter was about 14 months old, she was being particularly rough with the kitty - I saw that look in our cats eyes, and told Aislinn that if she did not stop pulling on the cats fur that she would get scratched (she understood this concept already after having seen mommy or daddy get a play scratch in rough housing with the kitty) - she kept on, and I let her. The cat clobbered her - jumped up and gave her a "whack" on either side of my daughters head, leaving one or two minor puncture wounds - no blood, but it definitely scared Aislinn. Needless to say, there has not been a single incident since - the kitty allows my daughter to pet her, scratch her, and rub her (even in her slightly aggressive way) and generally just leaves when it gets too rough for her. My daughter now also understands very clearly that being rough can hurt the kitty, and if she hurts her, the kitty might hurt her back - its all been part of the "gentle" process when dealing with aggression phase that toddlers go through (hitting hurts, biting hurts, kicking hurts, etc).
So, yes, it is possible for toddlers and cats to coexist, but you have to be okay with the fact that there might need to be a run in before your toddler understands that the kitty has feelings too.