I very much doubt that it can be done it in terms of all those parameters. You stand a chance for specific numerical values of the parameters, for example
With[{b1 = 1, b2 = 1, c1 = 1, c2 = 3, a1 = 0, a2 = -1},
InverseFunction[b1/(1 + E^(c1 (# - a1))) +
b2/(1 + E^(c2 (# - a2))) &]]
