The chances of rolling a 6 are 1 in 6 for every roll (same as any other particular number). The chances of not rolling a 6 are 5/6. We can write all the possible results of four rolls of the die by combining both these probabilities:
(1/6)^4, 4*(1/6)^3*(5/6), 6*(1/6)^2*(5/6)^2, 4*(1/6)*(5/6)^3, (5/6)^4
This means that there is one way of getting all sixes and no sixes; 4 ways of getting three sixes and not a six and getting three non-sixes and one six; six ways of getting 2 hits and two misses. The first three of these terms gives us the total probability of getting 4, 3 and 2 sixes when added together: 19/144=0.13194 or about 13.19%.