We have shown the possibility of non-Dicke superradiance for non-ideal magnon Bose-Einstein condensate (BEC) in a broadband frequency bath. Here, it is found that all the stored energy in the system of Bose-condensed magnons can be irradiated into a short pulse with a time delay caused by the strong frequency modulation of magnons due to direct inter-particle interactions in the Bose-condensed state. The last mechanism radically distinguishes this effect from the well-known Dicke superradiance of two-level atomic ensemble where the delay is connected with enhancement of the inter-atomic correlations due to exchange by virtual photons. In our case, the superradiance is the consequence of Bose-condensation in the coherent state where the particles are coupled by direct interaction. We have discussed the conditions for observation of this effect for Bose-condensed magnons in a solid-state sample with a spatial size smaller comparing with the wavelength of the emitted field. In general, we had shown that this kind of superradiance can proceed in samples with ferromagnetic type interaction. As for the antiferromagnetic ones, the effect of magnon superradiance takes place without delay.