Nondeterministic communication complexity of set disjointness?
$N^1(DISJ_n) \geq n$.

The fooling set technique actually provides lower bounds on the cover number. $C^1(f)$ is the minimum number of monochromatic rectangles needed to cover the $1$-inputs of $f$.  In the fooling set technique, you create a set of input pairs and show that each must lie in distinct monochromatic rectangles.  

In this case, look at the set of input pairs $\{(A,\bar{A}) : A \subseteq [n]\}$.  It is not too hard to show that $(A, \bar{A}), (B, \bar{B})$ must lie in different monochromatic rectangles if $A,B$ are distinct.  Hence, to cover the $1$-inputs, we need at least $2^n$ monochromatic rectangles.

Using $N^1(f) = \log_2(C^1(f))$, we get that $N^1(DISJ_n) \geq n$.

(This example is Example 1.23 from Kushilevitz/Nisan)