In this paper, we study the nonexistence of nontrivial global solutions for a system of two strongly coupled fractional differential equations. Each equation involves two fractional derivatives and a nonlinear source term. The fractional derivatives are of Caputo type of subfirst orders. The nonlinear sources are nonlocal in time. They have the form of a convolution of a polynomial of the state with a (possibly singular) kernel. The system under consideration is a generalization of many interesting special systems of equations whose solutions do not exist globally in time. We establish some criteria under which no nontrivial global solutions exist. Several integral inequalities and estimations are derived and the test function method is adopted. Special cases and examples are given to illustrate the results.