On a regular Shabbat, we read from 2 parshiyot when we have a communal Torah reading -- one parsha in Shacharit and the next at Mincha.
When there is a double parsha at Shacharit (such as Tazria-metzora) or some reason for there to be 2 sifrei Torah taken out in Shacharit, we read from 3 parshiyot.
If a double parsha falls out on a day when there is a separate reading from another sefer, and we take 2 sifrei Torah out, we read from 4 parshiyot, total. The same thing happens when we take 3 sifrei Torah out and read from a single parsha in each in Shacharit and then another in Mincha.
Is it possible in our current calendar to read from 5 Parshiyot? Can a day on which we take out 3 sifrei Torah coincide with a double parsha? Is there another permutation which would have leining from 5?