In the spectrum of the mixed boundary value problem for the Laplace operator in a finite, but long cylinder Ω^ℓ with curved ends, we detect a series of eigenvalues of the strange behavior as the length 2ℓ unlimitedly increases. In addition to usual hard-movable eigenvalues not leaving small neighborhoods of fixed points, we detect eigenvalues gliding down with a large speed along the real axis, but smoothly landing on the lower bound λ_† = Λ₁ of the continuous spectrum of the limit problem in the half-cylinder Π_± obtained by elongating Ω^ℓ in one of two directions. We give a complete description of the low-frequency range of the spectrum and show that each point in (Λ₁, Λ₂) is a blinking eigenvalue. Above the second threshold Λ₂ of the continuous spectrum of the problem in Π_±, the behavior of eigenvalues of the problem in the waveguide Ω^ℓ as ℓ → + ∞ is chaotic and their asymptotics can be constructed only in particular cases.