  Let $X$ be a Banach space. We prove that, for a large class of Banach or
quasi-Banach spaces $E$ of $X$-valued sequences, the sets $E-\bigcup
_{q\in\Gamma}\ell_{q}(X)$, where $\Gamma$ is any subset of $(0,\infty]$, and
$E-c_{0}(X)$ contain closed infinite-dimensional subspaces of $E$ (if
non-empty, of course). This result is applied in several particular cases and
it is also shown that the same technique can be used to improve a result on the
existence of spaces formed by norm-attaining linear operators.
