I'll corroborate what everyone else has said above. HOWEVER, I flew United out of IAD two weeks ago to the Caribbean. United has a new policy that forces you to check any fishing gear. It's outrageous that I had to check my two fly rods (which I managed to get into one hard case) each way when I saw folks with golf umbrellas, umbrella strollers, work posters, etc. carrying on. But the United agents on both ends were pretty adamant that I had to check my rods. I'm really glad that I checked the policy before I left the house, as I would have brought my Patagonia roll up rod case--which is great for carry-on, but would have probably resulted in multiple broken rods had I checked it. All of this is to say, check American's policy online before you assume that you can carry on your rods. If you do have to check rods, figure out a way to consolidate into a single rod carrier (I've had great luck with DIY rod carriers made out of PVC pipe from Home Depot).
Dave