Variational integrators are powerful tools for simulating constrained mechanical systems as well as computing optimal control strategies. While there are well-established methods to incorporate holonomic constraints in variational integrators, there have also been alternative suggestions that have appealing benefits in terms of computational efficiency. We compare the standard holonomic constraints method with what are called two-point constraint methods. In this paper both methods are tested on two different, relatively simple systems. The results reveal multiple problems with the twopoint constraints. They are found to be numerically unstable and to under some circumstances provide unreliable constraint force values. The conclusion is that two-point constraints are invalid and should not be used.