For countable discrete groups $\Gamma$, we consider $\epsilon$-representations of $\Gamma$ into unitary groups $U(n)$. These are unital maps $\rho: \Gamma \to U(n)$ such that $\|\rho(st) - \rho(s)\rho(t)\| < \epsilon$ for all $s, t \in \Gamma$. Kazhdan has shown that the surface groups of genus $> 1$ admit $\epsilon$-representations which are far from genuine representations in the point-norm topology. We exhibit new classes of hyperbolic groups $\Gamma$ which have the same instability features. More precisely, there exist a finite subset $F \subset \Gamma$ and $C > 0$ with the following property. For any $\epsilon > 0$ there is an $\epsilon$-representation $\rho: \Gamma \to U(n)$ such that for any representation $\pi: \Gamma \to U(n)$, $\max_{s \in F} \|\rho(s) - \pi(s)\| > C$.