  For a group G and a positive integer n write B_n(G) = {x \in G : |x^G | \le
n}. If s is a positive integer and w is a group word, say that G satisfies the
(n,s)-covering condition with respect to the word w if there exists a subset S
of G such that |S| \le s and all w-values of G are contained in B_n(G)S. In a
natural way, this condition emerged in the study of probabilistically nilpotent
groups of class two. In this paper we obtain the following results.
  Let w be a multilinear commutator word on k variables and let G be a group
satisfying the (n,s)-covering condition with respect to the word w. Then G has
a soluble subgroup T such that the index [G : T] and the derived length of T
are both (k,n,s)-bounded.
  Let G be a group satisfying the (n,s)-covering condition with respect to the
word \gamma_k. Then
  (1) \gamma_{2k-1}(G) has a subgroup $T$ such that the index [\gamma_{2k-1}(G)
: T] and |T'| are both (k,n,s)-bounded; and
  (2) G has a nilpotent subgroup U such that the index [G : U] and the
nilpotency class of U are both (k,n,s)-bounded.
