A critical analysis is made of a proposed mechanism to explain the ≲2- to 100-s delays observed in some flares between &ggr; rays due to several MeV protons and hard X rays due to ~100-keV electrons. The proposed mechanism is first-order Fermi accelerations of protons bouncing between two shocks formed by the electrons causing the hard X rays. For the maximum possible flux for stable beam propagation the shock formation time is found to be >2 s and the shock velocity ≲1800 km s-1. The mechanism as proposed produces protons which gain only ~3 MeV before they are transmitted through the shocks. The mechanism will work with additional pitch angle scattering, which also raises the possibility of multiple encounters with a single shock, a case not analyzed. For this reason the final energy gains of the proposed mechanism may be altered. The times during which this type of shock acceleration could be important and thus the range of observed delays which could be explained are given for a range of loop lengths and injected electron spectra. Alternate possibilities such as stochastic acceleration by hydromagnetic turbulence must be invoked to explain the shortest delays associated with this prompt proton acceleration.