  We study eight different gamma-ray burst (GRB) data sets to examine whether
current GRB measurements -- that probe a largely unexplored part of
cosmological redshift ($z$) space -- can be used to reliably constrain
cosmological model parameters. We use three Amati-correlation samples and five
Combo-correlation samples to simultaneously derive correlation and cosmological
model parameter constraints. The intrinsic dispersion of each GRB data set is
taken as a goodness measurement. We examine the consistency between the
cosmological bounds from GRBs with those determined from better-established
cosmological probes, such as baryonic acoustic oscillation (BAO) and Hubble
parameter $H(z)$ measurements. We use the Markov chain Monte Carlo method
implemented in \textsc{MontePython} to find best-fit correlation and
cosmological parameters, in six different cosmological models, for the eight
GRB samples, alone or in conjunction with BAO and $H(z)$ data. For the Amati
correlation case, we compile a data set of 118 bursts, the A118 sample, which
is the largest -- about half of the total Amati-correlation GRBs -- current
collection of GRBs suitable for constraining cosmological parameters. This
updated GRB compilation has the smallest intrinsic dispersion of the three
Amati-correlation GRB data sets we examined. We are unable to define a
collection of reliable bursts for current Combo-correlation GRB data.
Cosmological constraints determined from the A118 sample are consistent with --
but significantly weaker than -- those from BAO and $H(z)$ data. They also are
consistent with the spatially-flat $\Lambda$CDM model as well as with dynamical
dark energy models and non-spatially-flat models. Since GRBs probe a largely
unexplored region of $z$, it is well worth acquiring more and better-quality
burst data which will give a more definitive answer to the question of the
title.
