In a joint experimental and theoretical effort, we have studied dissociative electron attachment (DEA) to the CF3Br molecule at electron energies below 2 eV. Using two variants of the laser photoelectron attachment method with a thermal gas target (T(G) = 300 K), we measured the energy dependent yield for Br- formation over the range E = 3-1200 meV with resolutions of about 3 meV (E < 200 meV) and 35 meV. At the onsets for excitation of one and two quanta for the C-Br stretching mode nu3, downward cusps are detected. With reference to the recommended thermal (300 K) attachment rate coefficient k(A)(CF3Br) = 1.4 x 10(-8) cm3 s(-1), absolute cross sections have been determined for Br- formation. In addition, we studied Br- and (CF3Br)Br- formations with a seeded supersonic target beam (10% CF3Br in helium carrier gas, with a stagnation pressure of 1-4 bars and nozzle temperatures of 300 and 600 K) and found prominent structure in the anion yields due to cluster formation. Using the microwave pulse radiolysis swarm technique, allowing for controlled variation of the electron temperature by microwave heating, we studied the dependence of the absolute DEA rate coefficient on the mean electron energy E over the range of 0.04-2 eV at gas temperatures T(G) ranging from 173 to 600 K. For comparison with the experimental results, semiempirical resonance R-matrix calculations have been carried out. The input for the theory includes the known energetic and structural parameters of the neutral molecule and its anion; the parameters of the resonant anion curves are chosen with reference to the known thermal rate coefficient for the DEA process. For the gas temperature T(G) = 300 K, good overall agreement of the theoretical DEA cross section with the experimental results is observed; moreover, rate coefficients for Br- formation due to Rydberg electron transfer, calculated with both the experimental and the theoretical DEA cross sections, are found to agree with the previously reported absolute experimental values. At T(G) = 300 K, satisfactory agreement is also found between the calculated and experimental attachment rate coefficients for mean electron energies E = 0.04-2 eV. The strong increase of the measured rate coefficients with rising gas temperature, however, could be only partially recovered by the R-matrix results. The differences may result from the influence of thermal excitations of other vibrational modes not included in the theory.