  In this paper we focus on the interplay between the behaviour of the
Fr\"olicher spectral sequence and the existence of special Hermitian metrics on
the manifold, such as balanced, SKT or generalized Gauduchon. The study of
balanced metrics on nilmanifolds endowed with strongly non-nilpotent complex
structures allows us to provide infinite families of compact balanced manifolds
with Fr\"olicher spectral sequence not degenerating at the second page.
Moreover, this result is extended to non-degeneration at any arbitrary page.
Similar results are obtained for the Fr\"olicher spectral sequence of compact
generalized Gauduchon manifolds. We also find a compact SKT manifold whose
Fr\"olicher spectral sequence does not degenerate at the second page, thus
providing a counterexample to a conjecture by Popovici.
