Summary: We consider the eigenvalues \(\lambda_ 0(\varepsilon) < \lambda_ 1(\varepsilon) \leq \dots\) of the Neumann problem for the Laplace operator in a domain \(\Omega_ \varepsilon\) obtained by joining thin cylinders up to the lateral side of a thin plate (\(\Omega_ \varepsilon\) looks like a fine-teeth comb). The spectrum of the corresponding two- dimensional limit problem consists of normal eigenvalues and also of the accumulation points \(P_{m + 1}\) \((m = 0,1,\dots)\) which divide the eigenvalues into infinite sequences \(\{\mu^ m_ k\} \subset (P_ m,P_{m + 1})\). For any \(m,k \geq 0\) we find sequences \(\{\lambda_{n(\varepsilon,k,m)}(\varepsilon)\}\) converging to \(\mu^ m_ k\) as \(\varepsilon \to +0\).