Magnetostatic surface wave (MSSW) bright solitons in a ferrite-dielectric-metal (FDM) structure have been studied experimentally and numerically in the framework of the nonlinear Schrödinger equation. Attention was focused on the influence of the parametric instability on the soliton formation and propagation. We also discussed the contribution of the nonsolitary (dispersive wave) part of the MSSW pulse on the soliton propagation, to show that their mutual interference leads to the leveling off or to the appearance of some peaks in the MSSW pulse output versus the input amplitude. We have also shown that for MSSW pulses with rectangular shape, the linear pulse compression caused by an induced phase modulation of the input pulse must be taken into account. Experiments were performed on FDM microstrip structures loaded by a 14-μm-thick yttrium iron garnet film, separated from the ground metal by an air gap with thickness h1≈100 μm or h2≈200 μm. It was found experimentally for MSSW with wavelength λ≈h that the modulation instability leads to soliton formation for rectangular input pulses with duration τ less than the characteristic transient time t* needed for the onset of the parametric instability, while pulses with τ≥t* are mainly subjected to parametric instability. The measured threshold amplitudes for parametric and modulation instabilities are in agreement with the theoretical predictions. An influence of additional pumping in the form of both continuous-wave and pulsed signals on the soliton formation was studied. It was shown that an additional pumping signal with duration τ≥t*, and amplitude above the threshold of the parametric instability, suppressed the MSSW soliton. Numerical modeling of the pulsewidth dependence on the microwave power during the propagation in the FDM structure yields results that are in agreement with the experimental observations. Moreover, pulse narrowing due to the induced phase modulation of the input pulse was numerically predicted. All of these effects are in agreement with the experimental findings.