We consider the Robin Laplacian in the domains ω and ω^ϵ, ϵ > 0, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient a is large enough, the spectrum of the problem in ω is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain ω^ϵ is discrete. However, our results reveal the strange behaviour of the discrete spectrum as the blunting parameter ϵ tends to 0: we construct asymptotic forms of the eigenvalues and detect families of 'hardly movable' and 'plummeting' ones. The first type of the eigenvalues do not leave a small neighbourhood of a point for any small ϵ > 0 while the second ones move at a high rate O(| ln ϵ|) downwards along the real axis to -∞. At the same time, any point λ is a 'blinking eigenvalue', i.e., it belongs to the spectrum of the problem in ω^ϵ almost periodically in the | ln ϵ|-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.