We study n-dimensional complete intersections of "sufficiently high" multidegree in ℂP^n+k, n ≥ 3, with fixed number and, possibly, position of singularities. In the case of quadratic singularities, we give a topological description of such a complete intersection by decomposing it into a connected sum of special form. In this case, the diffeomorphism type of the variety is determined by its dimension, multidegree, and the number of singular points.