Do STDP synapses potentiate AND depress at the same time?
I can't make comments yet, so in the following I will assume that by 'STDP synapse' you mean the model of a synapse using the STDP rule.

The scenario that you are describing is a triplet experiment with a PRE-POST-PRE protocol and to answer your question: Yes, in this case the STDP rule predicts both, weakening and strengthening of the same synapse and given symmetric curves the net change would be zero.

In this paper in Figure 2 C,D they address this scenario (they report a non-zero net change but I guess it's due to asymmetric STDP curves). As they mention, the STDP rule is also symmetric in a different way in that it would predict the same net change in a POST-PRE-POST scenario because it simply adds up the two spike-pair events (PRE-POST and POST-PRE). Also in Figure 2 C,D is shown that experimental results differ from the STDP predictions because they are not the same for the PRE-POST-PRE and the POST-PRE-POST case. This discrepancy is actually the motivation of the more general triplet rule that is described in the reference.