  Suppose that $X$ is a Banach space. We will show that $X$ does not contain a
copy of $c_0$ if and only if for each series which is not unconditionally
convergent in $X$ respective sets coding all bounded subseries and
rearrangements are meager. We use Bessaga-Pe{\l}czy{\'n}ski $c_0-$Theorem and
concept of uniformly unconditionally bounded series. Moreover we prove similar
result for the ideal boundedness for a class of Baire ideals using Talagrand's
characterisation.
