The author studies the nonlinear two-point boundary value problem (*) \(x^{(m)}= f(t,x,x',\dots,\) \(x^{(m-1)})\), \(x^{(j)}(0)= A^ j\), \(j=0,1,\dots,m-1\), \(x^{(m-2)}=B\). Here \(m\geq 3\) is a given integer, \(x\) and \(f\) are given \(n\)-dimensional vector functions and \(A^ j\) and \(B\) are given \(n\)-dimensional constant vectors. He presents a new Nagumo condition, which has a simple form similar to the original Nagumo condition, and by means of which the above boundary value problem can be turned into boundary value problems for second order differential systems. Applying Nagumo's theorem and Schauder's fixed point theorem to this system the author obtains an existence theorem for solutions to \((*)\). Three nontrivial examples are discussed.