  One of the major goals of future Cosmic Microwave Background measurements is
the accurate determination of the effective number of neutrinos $N_{\rm eff}$.
Reaching an experimental sensitivity of $\Delta N_{\rm eff} = 0.013$ could
indeed falsify the presence of any non-standard relativistic particles at $95
\%$ c.l.. In this paper, we test how this future constraint can be affected by
the removal of two common assumptions: a negligible running of the inflationary
spectral index $n_{\rm run}$ and a precise determination of the neutron
lifetime $\tau_n$. We first show that the constraints on $N_{\rm eff}$ could be
significantly biased by the unaccounted presence of a running of the spectral
index. Considering the Stage-IV experiment, a negative running of ${\rm
d}n/{\rm d}\ln k= - 0.002$ could mimic a positive variation of $\Delta N_{\rm
eff} = 0.03$. Moreover, given the current discrepancies between experimental
measurements of the neutron lifetime $\tau_n$, we show that the assumption of a
conservative error of $\Delta\tau_n \sim 10$s could bring to a systematic error
of $\Delta N_{\rm eff} = 0.02$. Complementary cosmological constraints on the
running of the spectral index and a solution to the neutron lifetime
discrepancy are therefore needed for an accurate and reliable future CMB bound
of $N_{\rm eff}$ at percent level.
