A paradoxical behavior of the t-test under epsilon-contamination is pr
esented. The paradox consists in that under a fixed distribution of co
ntaminants an increasing of the probability of the appearance of a con
taminant may decrease the violation of the size of the test! A simple
explanation of the phenomenon is given. It is revealed which contamina
nts make the test conservative and which make it liberal: it appears t
hat, in spite of the established opinion, conservatism or liberalism o
f the test depends not so much on the tails of the contaminating distr
ibution as on where its support is located.