The dependent variables in a MANOVA should not be "too strongly correlated". But how strong a correlation is too strong? It would be interesting to get people's opinions on this issue. For instance, would you proceed with MANOVA in the following situations?
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Y1 and Y2 are correlated with $r=0.3$ and $p<0.005$
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Y1 and Y2 are correlated with $r=0.7$ and $p=0.049$
Update
Some representative quotes in response to @onestop:
"MANOVA works well in situations where there are moderate correlations between DVs" (course notes from San Francisco State Uni)
"The dependent variables are correlated which is appropriate for Manova" (United States EPA Stats Primer)
"The dependent variables should be related conceptually, and they should
be correlated with one another at a low to moderate level." (Course notes from Northern Arizona University)"DVs correlated from about .3 to about .7 are eligible" (Maxwell 2001, Journal of Consumer Psychology)
n.b. I'm not referring to the assumption that the intercorrelation between Y1 and Y2 should be the same across all levels of independent variables, simply to this apparent grey area about the actual magnitude of the intercorrelation.
Best Answer
There's no clear answer. The idea is that if you have a correlation that approaches 1 then you essentially have one variable and not multiple variables. So you could test against the hypotheses that r=1.00. With that said, the idea of MANOVA is to give you something more than a series of ANOVA tests. It helps you find a relationship with one test because you're able to lower your mean square error when combining dependent variables. It just won't help if you have highly correlated dependent variables.