  White dwarf stars have been used for decades as precise and accurate age
indicators. This work presents a test of the reliability of white dwarf total
ages when spectroscopic observations are available. We conduct follow-up
spectroscopy of 148 individual white dwarfs in widely separated
double-white-dwarf (WD+WD) binaries. We supplement the sample with 264
previously published white dwarf spectra, as well as 1292 high-confidence white
dwarf spectral types inferred from their Gaia XP spectra. We find that
spectroscopic fits to optical spectra do not provide noticeable improvement to
the age agreement among white dwarfs in wide WD+WD binaries. The median age
agreement is $\approx$$1.5\sigma$ for both photometrically and
spectroscopically determined total ages, for pairs of white dwarfs with each
having a total age uncertaintiy $<$ 20\%. For DA white dwarfs, we further find
that photometrically determined atmospheric parameters from spectral energy
distribution fitting give better total age agreement ($1.0\sigma$, 0.2 Gyr, or
14\% of the binary's average total age) compared to spectroscopically
determined parameters from Balmer-line fits (agreement of $1.5\sigma$, 0.3 Gyr,
or 28\% of binary's average total age). We find further evidence of a
significant merger fraction among wide WD+WD binaries: across multiple
spectroscopically identified samples, roughly 20\% are inconsistent with a
monotonically increasing initial-final mass relation. We recommend the
acquisition of an identification spectrum to ensure the correct atmospheric
models are used in photometric fits in order to determine the most accurate
total age of a white dwarf star.
