Let's say we have 7 samples of data and we wish to test whether there is a significant difference between the means of each sample. We could use t-tests, which would require 21 different t-tests. However, I've read that this isn't advisable as the risk of drawing the wrong conclusion (1 in 20) is increased. Instead, an ANOVA is advised.
But if 21 t-tests were performed using these data and because there are only 7 groups, some of the same data would be used in each test. So although we are calculating 21 different t-tests, we aren't using different data on 21 different occasions. So how does this situation increase the chance of drawing the wrong conclusion?