This paper deals with a coupled system of fourth-order parabolic inequalities ｜u｜ ≥ -△2u + ｜v｜q,｜v｜t ≥ -△2v + ｜u｜P in S =Rn × R+ with p,q ＞ 1,n ≥ 1.A FujitaLiouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n/4 ≤ max(p+1/pq-1,q+L/pq-1).Since the general maximum-comparison principle does not hold for the fourth-order problem,the authors use the test function method to get the global non-existence of nontrivial solutions.