  We perform a systematic study of the $^{56}$Ni mass ($M_{\rm Ni}$) of 27
stripped envelope supernovae (SESNe) by modeling their light-curve tails,
highlighting that use of ``Arnett's rule'' overestimates $M_{\rm Ni}$ for SESN
by a factor of $\sim$2. Recently, \citet{Khatami2019} presented a new model
relating the peak time ($t_{\rm p}$) and luminosity ($L_{\rm p}$) of a
radioactive-powered SN to its $M_{\rm Ni}$ that addresses several limitations
of Arnett-like models, but depends on a dimensionless parameter, $\beta$. Using
observed $t_{\rm p}$, $L_{\rm p}$, and tail-measured $M_{\rm Ni}$ values for 27
SESN, we observationally calibrate $\beta$ for the first time. Despite scatter,
we demonstrate that the model of \citet{Khatami2019} with
empirically-calibrated $\beta$ values provides significantly improved
measurements of $M_{\rm Ni}$ when only photospheric data is available. However,
these observationally-constrained $\beta$ values are systematically lower than
those inferred from numerical simulations, primarily because the observed
sample has significantly higher (0.2-0.4 dex) $L_{\rm p}$ for a given $M_{\rm
Ni}$. While effects due to composition, mixing, and asymmetry can increase
$L_{\rm p}$ current models cannot explain the systematically low $\beta$
values. However, the discrepancy can be alleviated if $\sim$7--50\% of $L_{\rm
p}$ for the observed sample originates from sources other than $^{56}$Ni.
Either shock cooling or magnetar spin-down could provide the requisite
luminosity. Finally, we find that even with our improved measurements, the
$M_{\rm Ni}$ values of SESN are still a factor of $\sim$3 larger than those of
hydrogen-rich Type II SN, indicating that these supernovae are inherently
different in terms of their progenitor initial mass distributions or explosion
mechanisms.
