Summary: We consider the following system of nonlinear third-order nonlocal boundary value problems (BVPs): \[ \begin{gathered} -u'''(t)= f(t,v(t), v'(t)),\quad t\in (0,1),\\ -v'''(t)= g(t,u(t), u'(t)),\quad t\in (0,1),\\ u(0)= 0,\quad au'(0)- bu''(0)= \alpha[u],\quad cu'(1)+ du''(1)= \beta[u],\\ v(0)= 0,\quad av'(0)- bv''(0)= \alpha[v],\quad cv'(1)+ dv''(1)= \beta[v],\end{gathered} \] where \(f,g\in C([0,1]\times \mathbb{R}^+\times \mathbb{R}^+, \mathbb{R}^+)\), \(\alpha[u]= \int^1_0 u(t)\,dA(t)\) and \(\beta[u]= \int^1_0 u(t)\,dB(t)\) are linear functionals on \(C[0,1]\) given by Riemann-Stieltjes integrals and are not necessarily positive functionals; \(a\), \(b\), \(c\), \(d\) are nonnegative constants with \(\rho:= ac+ ad+ bc>0\). By using the Guo-Krasnoselskii fixed point theorem, some sufficient conditions are obtained for the existence of at least one or two positive solutions and the nonexistence of positive solutions to the above problem. Two examples are also included to illustrate the main results.