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Let the weights of Abby, Bart, Cindy, and Damon be $a$, $b$, $c$, and $d$, respectively. We have the equations $$\begin{array}{rl}a+b& =260\\ b+c& =245\\ c+d& =270\end{array}$$ Subtracting the second equation from the first, we have $(a+b)-(b+c)=260-245\Rightarrow a-c=15$. Adding this last equation to the third given equation, we have $(a-c)+(c+d)=15+270\Rightarrow a+d=285$. Thus, Abby and Damon together weigh $\boxed{285}$ pounds.