The problem of bounding the reliability function of a multiple-access channel (MAC) is studied. An upper bound on the minimum Bhattacharyya distance between codeword pairs is derived. For a certain large class of two-user discrete memoryless (DM) MAC, a lower bound on the maximal probability of decoding error is derived as a consequence of the upper bound on Bhattacharyya distance. Further, an upper bound on the average probability of decoding error is studied. It is shown that the corresponding upper and lower bounds have a similar structure. Using a conjecture about the structure of the multi-user code, a tighter lower bound for the maximal probability of decoding error is derived and is shown to be tight at zero rates.