Moment tensors (MTs) of weak events are often calculated by a single agency (network), thus lacking independent validation. This article investigates how to increase reliability of the single-agency solutions through various multiple checks. It deals with the inversion of complete waveforms for six representative events (Mw 3.4–4.6) in the range of 0.08–0.15 Hz. Several three-station sets at near-regional distances (8–103 km) are used. The MTs are repeatedly calculated for two independent locations: from a regional and local network. The source depth is held fixed at the hypocenter and also grid searched to optimize the waveform match. Initially, large variations (instability) of the MT solution with the approach we used were found for some events. Later, the difficulties were understood and the solutions stabilized. It led to practical recommendations on how to detect or even avoid problematic MT solutions. First, avoid MT solutions for a single fixed depth (hypocenter). Second, optimize MT solutions by the depth grid search below epicenter (or better) for at least two alternative epicenters. Third, carefully analyze: (i) any rapid variation of the solution, (ii) major misfit of the first-motion polarities, (iii) low double-couple percentage (often correlating with [ii]), (iv) low values of the min/max eigenvalue ratio, and (v) departures of the MT-preferred depth from the location-derived depth. The MT-preferred depths, hence also the focal mechanism, may be misleading due to data problems (e.g., long-period disturbances, clipping) hidden in the band-pass waveforms. The location-derived depth may be especially wrong for very shallow earthquakes if the crustal model is inadequate in its shallow part and/or near stations are lacking. On the other hand, near stations (<20 km) should not be used in the waveform inversion together with distant stations; because of their large amplitudes, the inversion can be easily biased due to instrumental data errors and small location errors.