The bright P-3(1) - P-3(0) ([CI] 1-0) and P-3(2) - P-3(1) ([CI] 2-1) lines of atomic carbon are becoming more and more widely employed as tracers of the cold neutral gas in high-redshift galaxies. Here we present observations of these lines in the 11 galaxies of the set of Planck's Dusty GEMS, the brightest gravitationally lensed galaxies on the extragalactic submillimeter sky probed by the Planck satellite. We have [CI] 1-0 and [CI] 2-1 measurements for seven and eight of these galaxies, respectively, including four galaxies where both lines have been measured. We use our observations to constrain the gas excitation mechanism, excitation temperatures, optical depths, atomic carbon and molecular gas masses, and carbon abundances. Ratios of L-CI /L-FIR are similar to those found in the local universe, and suggest that the total cooling budget through atomic carbon has not significantly changed in the last 12 Gyr. Both lines are optically thin and trace 1-6 x 10(7) M-circle dot of atomic carbon. Carbon abundances, X-CI, are between 2.5 and 4 x 10(-5) , for an ultra-luminous infrared galaxy (ULIRG) CO-to-H-2 conversion factor of alpha(CO) = 0.8 M-circle dot/ [K km s(-1) pc(2) ]. Ratios of molecular gas masses derived from [CI] 1-0 and CO agree within the measurement uncertainties for five galaxies, and agree to better than a factor of two for another two with [CI] 1-0 measurements, after carefully taking CO excitation into account. This does not support the idea that intense, high-redshift starburst galaxies host large quantities of "CO-dark" gas. These results support the common assumptions underlying most molecular gas mass estimates made for massive, dusty, high-redshift starburst galaxies, although the good agreement between the masses obtained with both tracers cannot be taken as independent confirmation of either alpha(CO) or X-CI.