  We study the entanglement dynamics of quantum many-body systems at long times
and prove the following: (I) For any geometrically local Hamiltonian on a
lattice, starting from a random product state the entanglement entropy almost
never approaches the Page curve. (II) In a spin-glass model with random
all-to-all interactions, starting from any product state the average
entanglement entropy does not approach the Page curve. We also extend these
results to any unitary evolution with charge conservation and to the
Sachdev-Ye-Kitaev model. Our results highlight the difference between the
entanglement generated by (chaotic) Hamiltonian dynamics and that of random
states.
