In this paper, we investigate how the concept of cell conformance testing, developed within the ITU-T for a single class of traffic, could apply for connections with two types of traffic (namely with both CLP=0 and CLP=1 cells). For this purpose, we consider the Traffic Contract currently specified in ITU-T Recommendation I.371 (i.e., declaration of the peak cell rates of the CLP=0 and the aggregate CLP=0+1 cell streams). For the sake of completeness, we examine that considered earlier within the ITU-T and based upon the declaration of the peak cell rates of the CLP=0 and CLP=1 cell streams. To simplify the discussion and to be consistent with the current trend within the standardization bodies, we consider only connections for which Cell Loss Rate objectives have to be guaranteed for both CLP=0 and CLP=1 traffic. All cell conformance testing procedures, which may be envisaged for both Traffic Contracts, are examined and we show that most of them suffer from an undesirable phenomenon, namely measurement phasing, which may jeopardize their use. It turns out that only two procedures can be implemented for unambigous testing of cell conformance. The first one is defined in relation with the CLP=0 and CLP=1 Traffic Contract and the second one in relation with the CLP=0 and CLP=0+1 Traffic Contract. Even though the first cell conformance testing procedure allows a more accurate definition of the QoS for CLP=0 traffic, the second one based upon CLP=0 and CLP=0+1 controls run in parallel without coordination is the most in line with the current trend within the standardization bodies and the first implementation field trials.