  Recent cosmic microwave background (CMB) results from BOOMERANG, MAXIMA, and
DASI provide cosmological constraints on new physics that can be competitive
with those derived from Big Bang Nucleosynthesis (BBN). In particular, both CMB
and BBN can be used to place limits on models involving neutrino degeneracy and
additional relativistic degrees of freedom. However, for the case of the CMB,
these constraints are, in general, sensitive to the assumed priors. We examine
the CMB and BBN constraints on such models and study the sensitivity of ``new
physics" to the assumed priors. If we add a constraint on the age of the
universe ($t_0 \ga 11$ Gyr), then for models with a cosmological constant, the
range of baryon densities and neutrino degeneracy parameters allowed by the CMB
and BBN is fairly robust: $\eta_{10} = 6.0 \pm 0.6$, \deln $\la 6$, $\xi_e \la
0.3$. In the absence of new physics, models without a cosmological constant are
only marginally compatible with recent CMB observations (excluded at the 93%
confidence level).
