William Shanks was famous for having calculated Pi to 707 decimal places using Machin's theorem this took him many years, and was seen as a great feat. Soon after this had been completed De Morgan noticed that in near the end of this sequence there were very few 7's. This is written in his book called the budget of paradoxes which was published in 1872. In 1945 using a computer Ferguson found that infact the 528th digit was wrong and so therefore all the ones after were as well. Years of time wasted. The number of 7's did not in fact actually differ from the norm.
There is also much controversy surrounding Buffon's needle experiment (see calculating Pi section). Lots of people have tried to calculate Pi by throwing needles, and in 1901 Lazzerini said that be throwing the needle 34080 times he had obtained a value of pi of 3.1415929. It was later pointed out that if you decide when to stop the test you can get almost any value for Pi. If you stop it when you have a good value of Pi then the test will appear to be very good. This was very obviously shown up by Gridgeman.
Gridgeman was showing how silly it was to take this method seriously, so he took a needle of length 0.7857. Threw it twice hit a line once and said that this showed his value of Pi to be 3.1428. A good value. You can prove anything if you choose your facts carefully!
One man, Edmund Landau, even got sacked in a racist attack because Bieberbach didn't like his definition of Pi and said it was un-German like!
One state in America, Indiana, even tried to legally define Pi as 3. The bill was unanimously passed, but luckily the senate postponed the bill indefinitely.
Be it enacted by the General Assembly of the State of
Indiana: It has been found that a circular area is to the square on a line equal
to the quadrant of the circumference, as the area of an equilateral rectangle is
to the square of one side.
(Section I, House Bill No. 246, 1897)
Tenuous link: Pies
Picture taken from http://www.cds-computers.net/gg/archives/1002/1002pie.html