  We consider Big Bang nucleosynthesis and the cosmic microwave background when
both the neutrino temperature and neutrino number are allowed to vary from
their standard values. The neutrino temperature is assumed to differ from its
standard model value by a fixed factor from Big Bang nucleosynthesis up to the
present. In this scenario, the effective number of relativistic degrees of
freedom, $N_{\rm eff}^{\rm CMB}$, derived from observations of the cosmic
microwave background is not equal to the true number of neutrinos, $N_\nu$. We
determine the element abundances predicted by Big Bang nucleosynthesis as a
function of the neutrino number and temperature, converting the latter to the
equivalent value of $N_{\rm eff}^{\rm CMB}$. We find that a value of $N_{\rm
eff}^{\rm CMB} \approx 3$ can be made consistent with $N_\nu = 4$ with a
decrease in the neutrino temperature of $\sim 5\%$, while $N_\nu = 5$ is
excluded for any value of $N_{\rm eff}^{\rm CMB}$. No observationally-allowed
values for $N_{\rm eff}^{\rm CMB}$ and $N_\nu$ can solve the lithium problem.
