We present a general criterion based on the asymptotic behavior of basic sequences and Johnson-Schechtman technique, which guarantees large cardinality of the lattice of closed operator ideals in the algebra of bounded operators on a Banach space. The method yields $2^\mathfrak{c}$ closed operator ideals on a class of Lorentz sequence spaces, combinatorial spaces defined by compact families of finite subsets of integers, and spaces built on their basis - their $p$-convex versions and Baernstein spaces (extending the results of R.M.Causey - APB, N.J.Laustsen – J.Smith, A.Manoussakis - APB) and provides another approach to case of Rosenthal spaces, solved by W.B.Johnson and G.Schechtman.