"Precision" and "accuracy" are relative parameters and the interpretation depends on your demands; their exact values depend on your hardware and software.
RT measurements are finicky, even on a single pc simple basic USB polling latencies can add unacceptable latencies (8 ms or so) to RT estimates. Graphics latencies in visual tasks, or syncing issues in auditory-visual tasks can seriously mess up the stimulus timing. First-time calling of Matlab commands may add unacceptable latencies to the first trial. Etcetera etcetera. The Psychophysics Toolbox site has a wealth of information on these things.
Depending on why you wish to use a browser environment, but if you are attempting to do remote experiments over the internet, and assuming the software is stand-alone on the pc on the other end, you cannot control above-mentioned parameters at all. Further, if your pc is driving the experiment, things can get a lot worse, because a server or two will process the data in-between and there's the pc on the other end that you can't control.
But everything depends on the type of reaction time you are after. Are you interested in millisecond-range responses (standard react as-fast-as-you-can tasks), or do a few seconds more or less not matter (tasks where subjects get the time to make a cognitively evaluated choice)?
Also - if you are interested in latency differences (e.g. before and after drinking a few beers), then systematic latency deviations due to hardware issues do not matter anymore, as they are contained in both conditions and will be subtracted.
So it all depends on the experiment you are to use, the hardware that will be deployed and most of all, the type of response you are testing.