Extend your thinking: household appliances are usually connected in a parallel circuit. why do you think it might be a problem if too many appliances are turned on at once? (hint: current in a wire also produces heat.)

The equivalent resistance of several devices connected in parallel is given by

where  are the resistances of the various devices. We can see that every time we add a new device in parallel, the term  increases, therefore the equivalent resistance of the circuit  decreases.

But Ohm's law:

tells us that if the equivalent resistance decreases, the total current in the circuit increases. The power dissipated through the circuit (and so, the heat produced) depends on the square of the current:

therefore if there are too many devices connected in parallel, this can be a problem because there could be too much power dissipated (and too much heat) through the circuit.