Quote Originally Posted by Abdula
They didn't say it was a bloodline limit of the senju clan. Madara said that Hashirama was the tree element user from the senju clan, there isn't anything to suggest that it was a bloodline limit. Its plausible that Yamato could be half Senju but I doubt thats likely since all of Tsunade's relatives are supposed to be dead so I think she is the last of them.

The Uchiha don't seem like the type to mix blood, with all their Uchiha pride and such.

Yes they did. They just went through this in the anime, a few weeks ago. A bloodline limit is what happens when you mix two (or I guess more) elemental chakra types. And Yamato explained that the Wood Element was made up of water and earth elemental chakra. So the First's ability was a Bloodline limit.