I'm analysing my data for my master's dissertation and I'm having problems!

I have 3 "before" and "after" treatment groups (n=20 each) and I've applied paired samples t-test to the before and after data and found no significant differences between either of the groups at a p level of 0.05.

My vice-principal who is also in charge of the statistic module is telling us that we need to go on to do a one-way ANOVA analysis anyways (she's been wrong before and no one knows what they're talking about!!)

so I have 2 questions:

1) is an ANOVA really necessary since none of the groups show a statistically significant difference on t-tests?

2) so I did go ahead to do a one-way ANOVA test just to see, and despite the fact that my 3 paired t-tests are not significant, my ANOVA IS significant (p=0.005!)... How is this possible and what does it mean??

I'm very confused, please help!!