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In this article, study the existence of solutions for the second-order nonlinear coupled system of ordinary differential equations $$\displaylines{ u''(t)=f(t,v(t)),\quad t\in [0,1],\cr v''(t)=g(t,u(t)),\quad t\in [0,1], }$$ with nonlinear coupled boundary conditions $$\displaylines{ \phi(u(0),v(0),u(1),v(1),u'(0),v'(0))=(0,0), \cr \psi(u(0),v(0),u(1),v(1),u'(1),v'(1))=(0,0), }$$ where $f,g:[0,1]\times \mathbb{R}\to \mathbb{R}$ and $\phi,\psi:\mathbb{R}^6\to \mathbb{R}^2$ are continuous functions. Our main tools are coupled lower and upper solutions, Arzela-Ascoli theorem, and Schauder's fixed point theorem.