Can the percentages in a pie chart add up to more than $100\%$ and the data or analyses will still make sense (or be logical)? Let's say the percentage in a pie chart adds up to $100\%$ . Does this make statistical sense?
If the real numbers add up to 100%, and the difference is just round-off error, then it's okay, but I would include a note in the legend to explain.
If the numbers are not proportions of some whole, then you shouldn't use a pie chart. (And I'm not alone in thinking that you should find an alternative to a pie chart in any case.)
I know everyone has told you not to use a pie chart. I agree. But I think we should explain why. They are usually ok if there are two to three colours and there is only one of them AND you are speaking to people who only require a broad-strokes impression of the dominance of one area. If there are multiple sections one simply cannot make out the relative proportions of the chart and it becomes meaningless. So if you must use such a chart, I suggest you group it into three: 1-Biggest group, 2-Second biggest group, 3-All the other groups.
This will at least be readable. The other terrible, criminal (aarghh!, my eyes!) mistake to make is to use multiple pie charts with multiple colors. Most people's visual working memory is very poor, in contrast to their auditory working memory. Try closing your eyes and visualising the room around you in detail. Now try reciting a random nine digit number. You'll always have more success with the number (for all sorts of reasons). Using multiple pie charts means the viewer has to remember what the chart 2 slides ago looked like, or at the very least try and make a spatial comparison between 2 circular areas. Our brains are pretty bad at that too. If you want to compare changes across time, it is a much better idea to have TIME on at least one of your axes, with lines or blocks of color representing change. In this way a stacked bar chart can represent changing proportions much more naturally and sensibly than any pie.
In answer to the original question. No. Everybody else is being very nice. Think about it. If you could divide a pie up, and then put it back together again and have there be more than you started with, that would be a really useful party trick. There's a book about a guy who used to do stuff like that. Its name eludes me now.
- " You will eat, bye and bye, In that glorious land above the sky; Work and pray, live on hay, You'll get pie in the sky when you die" - The preacher and the slave, American folk song
If you start off with a pie chart, I have doubts about anything making sense :-). Pie charts are a terrible way to visualized data.
As @karl said, numbers can easily add up to 101% due to rounding error. I have a tee-shirt that says 2 + 2 = 5 (for extremely large values of 2).
But, to tell if the analysis makes sense, why not tell us what analysis you did?
As already noted, minor rounding errors can lead to values that are slightly greater than 100%.
Have you considered a bar chart instead of a pie chart? In my experience, bar charts make it easier for the reader to compare the sizes of your categories. This is especially so if you rank them in descending order. Alternately, you might also want to place similar categories adjacent to each other if those categories will make intuitive sense to your readers.