  In Functional Analysis, certain conclusions apply to sequences, but they
cannot be carried over when we consider nets. In fact, some nets, including
sequences, can behave unexpectedly. In this paper we are interested in
exploring the prevalence of these unusual nets in terms of linearity. Each
problem is approached with different methods, which have their own interest. As
our results are presented in the contexts of topological vector spaces and
normed spaces, they generalize or improve a few ones in the literature. We
study lineability properties of families of (1) nets that are weakly convergent
and unbounded, (2) nets that fail the Banach-Steinhaus theorem, (3) nets
indexed by a regular cardinal $\kappa$ that are weakly dense and
norm-unbounded, and finally (4) convergent series which have associated nets
that are divergent.
