Can ANOVA and t-test give different results?

Specifically, two group ANOVA vs unpaired t-test. My understanding was that those would always give the same p-value, but someone just told me that there could be rare cases where the ANOVA and the t-test would lead to different p-values. My questions: (a) is this true? and if yes, (b) under what conditions would this be more likely to occur? Thanks.

• Which t-test, exactly? There are several variants. Many people would automatically use a version that accommodates the possibility of unequal variances in the groups. That is likely to yield a p-value different from ANOVA, which assumes equal variances. – whuber Aug 28 '16 at 20:24
• If you compare equal variance t with equal variance anova on 2 groups they should give the same p-values; indeed, $t^2=F$. However if one uses say a Welch-Satterthwaite approach for dealing with unequal variance and the other does not (and either might) then they can differ. For the equal-variance case, I give an outline of the connection in my answer here (in the latter part) both doing some of the algebra and showing a specific numerical example (it doesn't do a complete derivation, however). – Glen_b -Reinstate Monica Aug 28 '16 at 22:55