|A or B| = |A| + |B| - |A and B|, extending to more sets by adding singles, subtracting pairs, adding triples, and so on.
Try the complement first. Questions phrased "at least one" are almost always easier as 1 minus "none". The probability at least one of four dice shows a six is 1 - (5/6)^4, not a four-term expansion.
Where it stops being practical. Terms double with each set. Beyond three or four overlapping conditions the formula is correct and hopeless under time pressure - if you are writing the fourth level, you have missed a symmetry.