I don't think it is neccessary to use almost any statistic if by that you mean mandatory If data is nested inside some factor regression (and I assume ANOVA which is a specialized form of regression) will generate incorrect results because two of the assumptions, homogenity of variance and independence of observations, won't be correct. For example, individuals inside a given schools will be systematically different than those outside it. For that the weighted least squares employed by multi-level methods is required (this assumes there is nesting which may or may not be true). From a practical perspective, explaining the variation by school or other parameter is highly useful as well. That is the random effects.
I am not sure what you mean by multivariate ANOVA (most ANOVA is multivariate if by that you mean two or more IV) but as far as I know all ANOVA requires equality of variance and indepenence of observation (well the later is not true in within subject ANOVA, but it does not seem like you are describing that).