  We present an extended zero-field muon spin relaxation (ZF-$\mu$SR) study of
overdoped Bi$_{2+x}$Sr$_{2-x}$CaCu$_2$O$_{8+\delta}$ (Bi2212) single crystals,
intended to elucidate the origin of weak quasistatic magnetism previously
detected by $\mu$SR in the superconducting and normal states of optimally-doped
and overdoped samples. New results on heavily-overdoped single crystals show a
similar monotonically decreasing ZF-$\mu$SR relaxation rate with increasing
temperature that persists above the pseudogap (PG) temperature $T^*$ and does
not evolve with hole doping ($p$). Additional measurements using an ultra-low
background apparatus confirm that this behavior is an intrinsic property of
Bi2212, which cannot be due to magnetic order associated with the PG phase.
Instead we show that the temperature-dependent relaxation rate is most likely
caused by structural changes that modify the contribution of the nuclear dipole
fields to the ZF-$\mu$SR signal. Our results for Bi2212 emphasize the
importance of not assuming the nuclear-dipole field contribution is independent
of temperature in ZF-$\mu$SR studies of high-temperature (high-$T_c$) cuprate
superconductors, and do not support a recent $\mu$SR study of
YBa$_2$Cu$_3$O$_{6+x}$ that claims to detect magnetic order in the PG phase.
