Speedy trial does not mean the duration of trial. It simply mean that government either has to present its evidence ( in speedy manner, statute usually set numbers of dates) in a trail or drop charges. Duration of any trial is depended upon complexity of the case. It takes as long as relevant evidence are being presented according to the criminal procedures of said jurisdiction.
The speedy trial statutes in New York state that a prosecutor has to state readiness for trial within a certain time after the arrest or arraignment. On a misdemeanor, that time is 90 days. On a felony, it's six months. (On a felony, the prosecutor cannot say he is ready until the case is indicted or reduced.)
Most prosecutors make a point of stating readiness on the first appearance in the court that is going to handle the case. They are trained to do so. After that, the time will only count against the prosecutor if subsequent delays are plainly requested by the prosecutor. As a layperson, you probably did not even notice this comment in court as it meant nothing to you.
Constitutional speedy trial (under the US Constitution) is a lot more vague.
If you do not have an attorney, you should get one. An experienced DWI attorney is best, especially if there are important issues with the case.
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Speedy trial is the period in which the DA must be ready to proceed. If you are charged with misdemeanor DWI, the period is 90 days. However, not all the time counts. If you lawyer either asked for an adjournment or consented to it, that time does not count. If you lawyer asked to make motions, the time does not count. As for the blood, there is a NY statute that says a driver is presumed to submit to a breath test. If you refused, its possible the DA or the police got a judge to sign a warrant allowing them to take blood. I am a former federal and State prosecutor and now handle criminal defense, including many DWI cases so feel free to contact me if you are looking to retain counsel.
I am a Nassau County practitioner and this DA is very difficult to say the least. However, whether you receive a speedy trial or not depends upon whether you were charged with a felony or a Misdemeanor. If its a Misdemeanor, then the DA has 90 days in which to bring you to trial. If a felony, then you double it to 180 days. However, you don't say whether you are represented by counsel now. If so, speak to them about the matter and see what they tell you. I suspect they've put the case off themselves trying to get Discovery or something else. If you still have problems, then hire another attorney.