The energetics of electromagnetic acceleration of ultrahigh-energy cosmic rays (UHECRs) is constrained both by confinement of a particle within an acceleration site and by radiative energy losses of the particle in the confining magnetic fields. We demonstrate that the detection of approximately 3 x 10(20) eV events is inconsistent with the hypothesis that compact cosmic accelerators with high magnetic fields can be the sources of UHECRs. This rules out the most popular candidates, namely spinning neutron stars, active galactic nuclei (AGNs). Galaxy clusters and, perhaps, AGN radio lobes and gamma-ray burst blast waves remain the only possible (although not very strong) candidates for UHECR acceleration sites. Our analysis places no limit on linear accelerators. With the data from the future Auger experiment one should be able to answer whether a conventional theory works or some new physics is required to explain the origin of UHECRs.