Answer:
Over ten. Specific dates where given in 1875, 1914, 1915, 1918, 1920, 1925, 1941, and 1975. Of course prior to the 1930's they were known as Bible Students. So technically Jehovah's Witnesses only did in 1941 and 1975. However JW's do accept the Bible Students as being the same organization. The reverse is not true however. In addition they have made predictions relating to a 'this generation'. While not giving a specific date, they have hinted at the mid nineties and more recently they extended that by a few years.
The Jehovah's Witness have published several "predictions" concerning Armageddon.
** 1914 Jehovah's Witnesses predicted that this year would be the end of the human affairs and the beginning of God's rulership on earth. All Jehovah's Witnesses at the time would have had to believe likewise because that is what was printed in the Watchtower publications. ( See 'The Time is at Hand 1907 pages 76-78)
** 1925 Judge Rutherford (2nd President of the Watchtower Society) said that "Millions [living at the time] would never die". The sweeping statement was later ( ie. after 1925) acknowledged by the Jehovah's Witnesses as based on an inaccurate understanding of bible prophecy and corrected.
1925 Jehovah's Witnesses believed was the time for Abraham etc to be resurrected on earth.
A number of other dates are worthy of note ...
**1874 & 1878 The Jehovah's Witness (along with other religious groups) speculated as to the significance of these years. Theses speculations were in retrospect (made AFTER the dates sited) so are not strictly speaking predictions.
http://en.wikipedia.org/wiki/Unfulfilled_religious_predictions
**1975 The Watchtower announced that this year marked the end of 6,000 years of human history but none of their literature ever stated categorically that this year would mark the "end of the world"rather it was stated that this was a"possibility". Instead it was very strongly implied. See the links below re 1975.
**The Generation of 1914. Jehovah's Witnesses said for about 50 years that Armageddon would come within the lifetime of people that saw the year 1914. This was stated as being 'Jesus's own words. In 1995 this was changed as that generation had all but ceased to exist. The latest prediction is that Armageddon will come within the lifetime of the generation of people that saw the generation of people that saw the year 1914. See link about generation teaching below.