Summary: We investigate the existence of solutions for a class of three-point boundary value problems for second order ordinary differential equations. We consider two types of nonlinearities, continuous nonlinearities and Carathéodory nonlinearities. The methods used in the first case rely on the topological transversality theorem, and in the second case we use the technique of cut-off functions. In both cases, a priori bounds on solutions play a key role in the analysis.