The photoresponse in the electrical conductivity of a single-walled carbon nanotube (SWNT) film is dramatically enhanced when the nanotube film is suspended in vacuum. We show here that the change in conductivity is bolometric (caused by heating of the SWNT network). Electron-phonon interactions lead to ultrafast relaxation of the photoexcited carriers, and the energy of the incident infrared (IR) radiation is efficiently transferred to the crystal lattice. It is not the presence of photoexcited holes and electrons, but a rise in temperature, that results in a change in resistance; thus, photoconductivity experiments cannot be used to support the band picture over the exciton model of excited states in carbon nanotubes. The photoresponse of suspended SWNT films is sufficiently high that they may function as the sensitive element of an IR bolometric detector.