Are there any bacteria, in our environment, able to break down nitrate under aerobic conditions?
My initial reaction upon reading this question was to think that the answer was: no, denitrification is a process in which nitrate is used in the absence of oxygen as an alternative electron acceptor. However I decided to check the literature, and I have discovered that there is, in fact, a lot of new evidence that many bacterial species may be able to use both of these electron acceptors simultaneously. This article is a good place to start if you want to find out more:


  Chen &amp; Strous (2013) Denitrification and aerobic respiration, hybrid electron transport chains and co-evolution. BBA Bioenergetics 1827: 136-144  


This is a review which summarises the area from the point of view of bioenergetics, but most importantly for this question, includes a Table summarising the experimental evidence for the phenomenon.  Referring to this Table, they state:


  It appears that in most cases, the evidence presented is not sufficient to support the conclusion that denitrification really proceeded in the presence of excess oxygen. It should be noted that the aerobic denitrification rates were much lower than the anaerobic denitrification rates measured for the same strain while aerobic respiration rates remain high. This implies that the major electron flow in the respiratory chain is towards complex IV rather than the NOx reductases, and that aerobic respiration is still the preferred pathway for growth. Apparently aerobic denitrification can be an auxiliary pathway next to aerobic respiration.


The last sentence answers the OP question.