  Let $G$ be a finite group, and $c$ an element of $\mathbb{Z}\cup \{\infty\}$.
A subgroup $H$ of $G$ is said to be {\it $c$-nilpotent} if it is nilpotent, and
has nilpotency class at most $c$. A subset $X$ of $G$ is said to be {\it
non-$c$-nilpotent} if it contains no two elements $x$ and $y$ such that the
subgroup $< x,y>$ is $c$-nilpotent. In this paper we study the quantity
$\omega_c(G)$, defined to be the size of the largest non-$c$-nilpotent subset
of $L$.
  In the case that $L$ is a finite group of Lie type, we identify covers of $L$
by $c$-nilpotent subgroups, and we use these covers to construct large
non-$c$-nilpotent sets in $L$. We prove that for groups $L$ of fixed rank $r$,
there exist constants $D_r$ and $E_r$ such that $D_r N \leq \omega_\infty(L)
\leq E_r N$, where $N$ is the number of maximal tori in $L$.
  In the case of groups $L$ with twisted rank 1, we provide exact formulae for
$\omega_c(L)$ for all $c\in\mathbb{Z}\cup \{\infty\}$. If we write $q$ for the
level of the Frobenius endomorphism associated with $L$ and assume that $q>5$,
then $\omega_\infty(G)$ may be expressed as a polynomial in $q$ with
coefficients in $\{0,1\}$.
