Yes. In the same way that flipping a fair coin (with equal probabilities of getting heads of tails) eight times in a row is likely to come up all heads 1/256 times, or all tails 1/256 times. The psychological perception of a sequence with a run of 8 heads or 8 tails is that it is so unlikely as to never occur at all; whereas we mathematicians see the likelihood of a run of 8 in 8 flips as occuring with 2/256 or just under 1% of the time.
The opposite error is true, and also occurs with some frequency in biomedical experiments and medical experiments. The standard for accepting a result in a clinical or medical trial is for $p<0.05$: that there is less than a 5% probability that the results occured by chance. Thus, one in twenty times, it is possible that a random occurence or set of occurences will be perceived or accepted as being statistically valid when it is not.