  We present experimental measurements of the steady-state ion number in a
linear Paul trap (LPT) as a function of the ion-loading rate. These
measurements, taken with (a) constant Paul trap stability parameter $q$, (b)
constant radio-frequency (rf) amplitude, or (c) constant rf frequency, show
nonlinear behavior. At the loading rates achieved in this experiment, a plot of
the steady-state ion number as a function of loading rate has two regions: a
monotonic rise (region I) followed by a plateau (region II). Also described are
simulations and analytical theory which match the experimental results. Region
I is caused by rf heating and is fundamentally due to the time dependence of
the rf Paul-trap forces. We show that the time-independent pseudopotential,
frequently used in the analytical investigation of trapping experiments, cannot
explain region I, but explains the plateau in region II and can be used to
predict the steady-state ion number in that region. An important feature of our
experimental LPT is the existence of a radial cut-off $\hat R_{\rm cut}$ that
limits the ion capacity of our LPT and features prominently in the analytical
and numerical analysis of our LPT-loading results. We explain the dynamical
origin of $\hat R_{\rm cut}$ and relate it to the chaos border of the fractal
of non-escaping trajectories in our LPT. We also present an improved model of
LPT ion-loading as a function of time.
