I agree with the majority of sentiment in here. In response to a TA, by all means ask the ATC or other pilot whats going on and sort something out if need be. But when you get an RA, I as an ATC will be basing my response on the assumption that both aircraft are responding as commanded by TCAS and as such I will not give any instructions that may contradict the TCAS. Too many cooks spoil the broth, and in this case a couple of TCAS units chatting to each other and working things out is more than enough cooks.
I understand Ferris, about all the limitations of TCAS, and I have had my fair share of false RA's making aircraft under my control climb or descend unneccessarily, but on balance I think with the time constraints we are talking about, and the consequences of not getting it right, I say follow the TCAS and follow it properly and swiftly.
I do not believe an airspace system should ever be designed with it as one of the safety features as it is and for the foreseeeable future will be, a net that saves the day if everything else goes wrong. But thanks to the wisdom of Smith and Anderson, we have a system that will on a regular basis rely on TCAS (and most likely in cases where noone has done anything wrong!), so we all better make sure we are all on the same page when it comes to responding always to TCAS or not. So whats it gonna be???