We investigate quenching processes which contribute to the roll-off in quantum efficiency of phosphorescent organic light-emitting diodes (OLED's) at high brightness: triplet-triplet annihilation, energy transfer to charged molecules (polarons), and dissociation of excitons into free charge carriers. The investigated OLED's comprise a host-guest system as emission layer within a state-of-the-art OLED structure---i.e., a five-layer device including doped transport and thin charge carrier and exciton blocking layers. In a red phosphorescent device, $N,{N}^{\ensuremath{'}}$-di(naphthalen-2-yl)- $N,{N}^{\ensuremath{'}}$-diphenyl-benzidine is used as matrix and tris(1-phenylisoquinoline) iridium $[\mathrm{Ir}(\mathrm{piq}{)}_{3}]$ as emitter molecule. This structure is compared to a green phosphorescent OLED with a host-guest system comprising the matrix 4,${4}^{\ensuremath{'}}$,${4}^{\ensuremath{''}}$-tris ($N$-carbazolyl)-triphenylamine and the well-known triplet emitter fac-tris(2-phenylpyridine) iridium $[\mathrm{Ir}(\mathrm{ppy}{)}_{3}]$. The triplet-triplet annihilation is characterized by the rate constant ${k}_{TT}$ which is determined by time-resolved photoluminescence experiments. To investigate triplet-polaron quenching, unipolar devices were prepared. A certain exciton density, created by continuous-wave illumination, is analyzed as a function of current density flowing through the device. This delivers the corresponding rate constant ${k}_{P}$. Field-induced quenching is not observed under typical OLED operation conditions. The experimental data are implemented in an analytical model taking in account both triplet-triplet annihilation and triplet-polaron quenching. It shows that both processes strongly influence the OLED performance. Compared to the red $\mathrm{Ir}(\mathrm{piq}{)}_{3}$ OLED, the green $\mathrm{Ir}(\mathrm{ppy}{)}_{3}$ device shows a stronger efficiency roll-off which is mainly due to a longer phosphorescent lifetime $\ensuremath{\tau}$ and a thinner exciton formation zone $w$.