Can we say anything about the complement of a regular language?
Complementing an NFA may involve an exponential blowup.
While there are concrete examples for this, it can also be deduced (in a way) by the fact that NFA universality is PSPACE complete. Indeed, if complementing an NFA could be done in polynomial time, then universality would also be possible in polynomial time.
This is not exact, since complemetation could also be only sub-exponential, but not polynomial.
As for a concrete example: Let $p_1,p_2,\dots$ be an enumeration of the prime numbers.
Consider the language: $L_k=\{1^n: \exists 1\le i\le k\ \text{ s.t. }\ p_i \not\mid n\}$. It is easy to construct an NFA for $L_k$ with $\sum_{i=1}^kp_i$ states.
However, it is also not hard to prove that a minimal NFA for $\overline{L_k}$ needs $\prod_{i=1}^k p_i$ states, which is exponential in $\sum_{i=1}^kp_i$.