Since you are not an electrician, you won't get any help on anything that involves actually working on this equipment, changing things out, or measurements that involve contact with the wiring itself. But there is still lots that we can help you with.
First thing I'd suggest is that you get a picture of the switchgear and the front of the breaker, and after it trips get another picture before it gets reset. Post the pictures here for feedback. If you can figure out if the system is 120/240V or 208/120V, that would be helpful.
The point that
@Jraef made in post #8 is important: large breakers often have 'ground fault' sensing built in. This is the big brother of the sort of sensing used in GFCIs, but is intended for protecting electrical systems from large faults rather than protecting people from shocks. This generally isn't required for a residential building, but _might_ be present anyway. The problem is this: often the ground fault system of the main breaker is often more sensitive than the overcurrent tripping of a branch circuit, so the main trips instead of the branch circuit. IMHO this is the first issue to rule out.
Next check with the power company to see if you can get any 'demand' readings for the service. It might be that for short periods of time your apartment buildings are actually drawing excess power, and the main breaker is tripping exactly as it should.
You may need to get your electrician to install some sort of power monitor; something that measures current, voltage, and ground fault current, and records what is going on. Also get a video camera pointed at the breaker!
-Jonathan