According to the New York State Department of Vehicles site, motor vehicle points are counted from the date of your traffic violations, not from the date of your traffic convictions.  A traffic conviction though is required for the points to appear on your driving record.

Eighteen months after the date of the violation, the points for that violation will be removed from your point total by the DMV.  The conviction itself though remains on your record for the appropriate length of time according to NY statutes.

A moving violation conviction or an accident in NY normally remains on a driver's record during the year of the conviction occurred and for the following 3 calendar years.  For this the DMV uses the year that the conviction happened, not the year of that the violation occurred.

The DMV removes a conviction or an accident from a NYS driver's record on January 1 of the forth year after the conviction.  Unless it is a serious conviction or accident of a serious type, those convictions can remain on a NYS driving history for more than 10 years.

It would thus appear that if you received a conviction now for a 6 year old speeding ticket that the points that would have been assessed would now have dropped off.  I would suggest checking to make certain this is true for your particular situation by contacting the NYS DMV. 

If you are convicted of the speeding offense, the violation conviction would appear on your driving record for the normal period of time.  Your insurance company could thus see the violation so it could affect your rates.  If it does, come to us for an affordable New York auto insurance quote.