We investigate configurations of conics in the projective plane which have the property that the number of tacnodes is equal or close to the upper bound obtained from the Miyaoka-Yau inequality. We show that for 5 conics there are exactly 3 configurations, including 2 new ones, achieving the maximum 17 tacnodes, and for 6 conics the maximum number of tacnodes is 22, which together with previous results implies that the Miyaoka-Yau bound can never be achieved.