The flat-spectrum radio quasar PKS 1222+216 (4C+21.35, z = 0.432) was detected in the very high energy γ-ray band by MAGIC during a highly active γ-ray phase following an alert by the Large Area Telescope (LAT) onboard Fermi. Its relatively hard spectrum (70-400 GeV photon index Γ = 2.7 ± 0.3) without a cut off, together with its observed variability on a timescale of ~10 min challenges standard emission models. In particular, if the emission originates in a portion of the relativistic jet located inside the broad line region (BLR), severe absorption of γ rays above a few tens of GeV is expected to be caused by the γγ → e ± process. These observations therefore imply that there is a very compact (Rb ~ 5 × 1014 cm) and rapidly moving blob located far beyond the BLR radius (to avoid the gamma-ray absorption through pair production) that is responsible for the rapidly varying high energy flux. However, the long-term (day-week) coherent evolution of the GeV flux recorded by LAT indicates that there could also be a substantial contribution from another, larger emission region. We model the spectral energy distribution of PKS 1222+216 during the epoch of the MAGIC detection assuming three different scenarios, namely: (1) a one-zone model considering only the emission from a compact blob outside the BLR; (2) a two-zone model consisting of a compact blob plus an emitting region encompassing the whole jet cross-section located outside the BLR; and (3) a two-zone model with the jet emitting region inside the BLR. In all cases we find that the high-energy emission from the compact blob is dominated by the inverse Compton scattering of the infrared thermal radiation of the dusty torus. Furthermore, both regions are matter-dominated, with the Poynting flux providing a negligible contribution to the total jet power. These results do not support models in which the compact blob is the result of reconnection events inside the jet or "needles" of high-energy electrons accelerated close to the BH. The observational framework and our radiative models might instead be compatible with scenarios in which the jet is re-collimated and focussed at large distances from the central BH.