The REML procedure is a 'search algorithm' or 'numerical optimization.' These methods are used when a closed-form solution does not exist. Such methods are generally not guaranteed to find the global optimum, but they often take steps to minimize converging to a local optimum. Furthermore, the finite precision of the numerical routines and processor can lead to various errors (not an error message, but a departure from the true value). Lastly, most methods quit when one of several convergence criteria is met. That is to say, it is not an exhaustive search.
Lack of convergence might produce 'good enough' results, but it is best not to trust the results in such a case. Failure to converge can be caused by the nature of the data or the convergence criteria are too strict.
