In this Letter we report on the investigation of the upconversion emission of single NaYF(4) nanocrystals codoped with Yb(3+) and Er(3+). Single nanocrystals on a coverslip are excited with continuous wave laser light at 973 nm in a confocal setup and the upconversion fluorescence is analyzed with a spectrometer. With the help of an atomic force microscope the size of the nanocrystals is simultaneously determined. A strong size-dependence of the spectral properties of the upconversion signal of individual nanocrystals is observed. We attribute this to a differing number of available phonons in the individual crystals for multiphonon relaxation processes, depending on their size. We believe that this result provides a new strategy in the synthesis of upconversion nanoparticles with different spectral properties by changing only their size as it is well-known from the case of semiconductor quantum dots.