Is there a state mandated cap for electricity use in Spain?
I'm from Spain, and yes, your father is right there is a cap in terms of contracted power with the electrical company. In the contract you sign with the electrical company it's stated the maximum power you need (at the same time). For example, in my case I've a maximum contracted power of 5.75KW (with this power, I've never had problems having at the same time running the AC, the oven, the fridge, a computer, some lights, and the water heater). And to my knowledge, particulars (that's not companies) can contract a power up to 9.7KW.

So the bill depends on two factors, the amount of energy you've been using during a period of time (variable), and the maximum energy you've contracted (fixed amount). An easy way to slightly reduce the bill is contracting a lower maximum energy, but in that case you may face power cuts if you surpass this maximum in a given moment. If that's your father problem, he should call his power supplier and ask them to increase the maximum energy (they usually charge for this kind of changes, and moreover the bill will be proportionally higher). He can know this maximum looking at a bill (looking at "Potencia").

But it's strange what happens to you father, mainly taking in mind that it's a new house. New houses projects usually include a study of average electrical power needs, so that you contract the power according to the study to avoid cuts. So I'd not discard an internal wiring problem.