  Given a sequence of curves on a surface, we provide conditions which ensure
that (1) the sequence is an infinite quasi-geodesic in the curve complex, (2)
the limit in the Gromov boundary is represented by a nonuniquely ergodic ending
lamination, and (3) the sequence divides into a finite set of subsequences,
each of which projectively converges to one of the ergodic measures on the
ending lamination. The conditions are sufficiently robust, allowing us to
construct sequences on a closed surface of genus $g$ for which the space of
measures has the maximal dimension $3g-3$, for example.
  We also study the limit sets in the Thurston boundary of Teichm\"uller
geodesic rays defined by quadratic differentials whose vertical foliations are
obtained from the constructions mentioned above. We prove that such examples
exist for which the limit is a cycle in the $1$-skeleton of the simplex of
projective classes of measures visiting every vertex.
