We perform a systematic study of the possible loosely bound states composed of two charmed baryons or a charmed baryon and an anticharmed baryon within the framework of the one-boson-exchange model. We consider not only the pi exchange but also the eta, rho, omega, phi, and sigma exchanges. The S - D mixing effects for the spin triplets are also taken into account. With the derived effective potentials, we calculate the binding energies and rms radii for the systems Lambda(c)Lambda(c)((Lambda) over bar (c)), Xi(c)Xi(c)((Xi) over bar (c)), Sigma(c)Sigma(c)((Sigma) over bar (c)), Xi(c)'Xi(c)'((Xi) over bar (c)'), and Omega(c)Omega(c)(Omega) over bar (c). Our numerical results indicate that ( 1) the H-dibaryonlike state Lambda(c)Lambda(c) does not exist and (2) there may exist four loosely bound deuteronlike states for the Xi(c)Xi(c) and Xi(c)'Xi(c)' systems with small binding energies and large rms radii.