The phylogenetic bootstrap is the most commonly used method for assessing statistical confidence in estimated phylogenies by non-Bayesian methods such as maximum parsimony and maximum likelihood (ML).  It is observed that bootstrap support tends to be high in large genomic datasets whether or not the inferred trees and clades are correct. Here we study the asymptotic behavior of bootstrap support for the ML tree in large datasets when the competing phylogenetic trees are equally right or equally wrong.  We consider phylogenetic reconstruction as a problem of statistical model selection when the compared models are nonnested and misspecified. The bootstrap is found to have qualitatively different dynamics from Bayesian inference, and does not exhibit the polarized behavior of posterior model probabilities, consistent with the empirical observation that the bootstrap is more conservative than Bayesian probabilities. Nevertheless bootstrap support similarly shows fluctuations among large datasets, with no convergence to a point value, when the compared models are equally right or equally wrong. Thus in large datasets strong support for wrong trees or models is likely to occur. Our analysis provides a partial explanation for the high bootstrap support values for incorrect clades observed in empirical data analysis.