It is known that there exist signals in Paley-Wiener space $PW_\pi^1$ of bandlimited signals with absolutely integrable Fourier transform, for which the peak value of the Shannon sampling series diverges unboundedly. In this paper we analyze the structure of the set of signals which lead to strong divergence. Strong divergence is closely linked to the existence of adaptive methods. We prove that there exists an infinite dimensional closed subspace of $PW_\pi^1$, all signals of which, except the zero signal, lead to strong divergence of the peak value of the Shannon sampling series.