We investigate the number of maximal independent set queries required to reconstruct the edges of a hidden graph. We show that randomised adaptive algorithms need at least $Ω(Δ^2 \log(n / Δ) / \log Δ)$ queries to reconstruct $n$-vertex graphs of maximum degree $Δ$ with success probability at least $1/2$, and we further improve this lower bound to $Ω(Δ^2 \log(n / Δ))$ for randomised non-adaptive algorithms. We also prove that deterministic non-adaptive algorithms require at least $Ω(Δ^3 \log n / \log Δ)$ queries. This improves bounds of Konrad, O'Sullivan, and Traistaru, and answers one of their questions. The proof of the lower bound for deterministic non-adaptive algorithms relies on a connection to cover-free families, for which we also improve known bounds.