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If they can't, the lights will go out.
Normal, modern apartment buildings will typically have at least 15a plugs and breakers as will a house. That's pretty much a given based on building codes. (Unless you live in an area without building codes and they do exist!). With older construction all bets are off and you have to do a little more leg work to verify what's going on.
Normal, modern commercial construction like offices will have 20a plugs and breakers.
For your reference, a 15a plug/circuit should normally support up to around 1200w load without too much problem--but of course your mileage may vary based on all the variables involved including the quality of the wiring in the walls.
Tungsten lighting is extremely power-hungry. On a 15-amp circuit you could use maybe two 650w lights, as long as nothing else is powered on that circuit.
For shorthand, a 15-amp circuit should support 1500 watts. A 20-amp circuit should be able to support 2000 watts. There's some margin of error there, but it's a shorthand rule that should give you some ideas.
Frequently bathrooms or kitchens will be on their own circuits, as will laundry rooms, so you might have success if you just route the lights around the apartment to various locations (i.e., plug one into the kitchen, one into the living room, one into the bathroom). That's probably being excessively overprotective, but it should make sure that you have no problems working with a small Arri kit.
Turn off the coffee pot. That and the toaster/microwave can ruin your day if on the same circuit.
What type of damage is being caused if the lights go out?