  The sensitivity of pulsar timing arrays to gravitational waves is, at some
level, limited by timing noise. Red timing noise - the stochastic wandering of
pulse arrival times with a red spectrum - is prevalent in slow-spinning pulsars
and has been identified in many millisecond pulsars. Phenomenological models of
timing noise, such as from superfluid turbulence, suggest that the timing noise
spectrum plateaus below some critical frequency, $f_c$, potentially aiding the
hunt for gravitational waves. We examine this effect for individual pulsars by
calculating minimum observation times, $T_{\rm min}(f_c)$, over which the
gravitational wave signal becomes larger than the timing noise plateau. We do
this in two ways: 1) in a model-independent manner, and 2) by using the
superfluid turbulence model for timing noise as an example to illustrate how
neutron star parameters can be constrained. We show that the superfluid
turbulence model can reproduce the data qualitatively from a number of pulsars
observed as part of the Parkes Pulsar Timing Array. We further show how a value
of $f_c$, derived either through observations or theory, can be related to
$T_{\rm min}$. This provides a diagnostic whereby the usefulness of timing
array pulsars for gravitational-wave detection can be quantified.
