  In this paper, we study the cones of higher codimension (pseudo)effective
cycles on point blow-ups of projective space. We determine bounds on the number
of points for which these cones are generated by the classes of linear cycles,
and for which these cones are finitely generated. Surprisingly, we discover
that for (very) general points, the higher codimension cones behave better than
the cones of divisors. For example, for the blow-up $X_r^n$ of $\mathbb P^n$,
$n>4$, at $r$ very general points, the cone of divisors is not finitely
generated as soon as $r> n+3$, whereas the cone of curves is generated by the
classes of lines if $r \leq 2^n$. In fact, if $X_r^n$ is a Mori Dream Space
then all the effective cones of cycles on $X_r^n$ are finitely generated.
