jmc

First, we use difference of squares on $13^4-11^4=(13^2{)}^2-(11^2{)}^2$ to get $13^4-11^4=(13^2+11^2)(13^2-11^2)$. Using difference of squares again and simplifying, we get $(169+121)(13+11)(13-11)=290\cdot 24\cdot 2=(2\cdot 8\cdot 2)\cdot (3\cdot 145)$. Realizing that we don't need the right-hand side because it doesn't contain any factor of 2, we see that the greatest power of $2$ that is a divisor $13^4-11^4$ is $\boxed{\text{mbox}32}$.